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This pdf includes the following topics:-

The volume of Prisms: Using a Formula

Surface Area of Prisms

The volume of Rectangular Prisms

Examples and many more.

The volume of Prisms: Using a Formula

Surface Area of Prisms

The volume of Rectangular Prisms

Examples and many more.

1.
Volume of Prisms: Using a Formula Pathway

OPEN-ENDED

1

There are a variety of ways to create prisms and to figure out You will need

their volume.

• linking cubes

Part A • a camera (optional)

• pattern blocks

• Use linking cubes to make 3 different non-rectangular one-layer • centimetre cubes

bases. Then stack identical shapes on top of each base to build • a ruler

prisms. Take photos of your prisms if a camera is available. • a calculator

• Sketch the base of each prism you made.

Base of Prism 1: Base of Prism 2: Base of Prism 3: Re

emember

member

e.g., e.g., e.g.,

• Volume is the

amount of space a

shape takes up.

It is measured in

cubic units.

• Volume of a

rectangular prism 5

• Write the area of each base, in square units, and the number

area of base 3 height

of layers in each of your prisms.

h

Prism 1: Prism 2: Prism 3:

e.g., 5 square e.g., 5 square e.g., 5 square

units, 3 layers units, 4 layers units, 5 layers w

l

• How does the volume of each prism relate to the area of the base

and the number of layers?

e.g., The volume is the number of cubes, which is same as the product

of the base area and the height.

• Why does that make sense?

e.g., Each layer forms another equal group, and you multiply when

there are equal groups.

353

Copyright © 2012 by Nelson Education Ltd. Leaps and Bounds Volume of Prisms: Using a Formula, Pathway 1

OPEN-ENDED

1

There are a variety of ways to create prisms and to figure out You will need

their volume.

• linking cubes

Part A • a camera (optional)

• pattern blocks

• Use linking cubes to make 3 different non-rectangular one-layer • centimetre cubes

bases. Then stack identical shapes on top of each base to build • a ruler

prisms. Take photos of your prisms if a camera is available. • a calculator

• Sketch the base of each prism you made.

Base of Prism 1: Base of Prism 2: Base of Prism 3: Re

emember

member

e.g., e.g., e.g.,

• Volume is the

amount of space a

shape takes up.

It is measured in

cubic units.

• Volume of a

rectangular prism 5

• Write the area of each base, in square units, and the number

area of base 3 height

of layers in each of your prisms.

h

Prism 1: Prism 2: Prism 3:

e.g., 5 square e.g., 5 square e.g., 5 square

units, 3 layers units, 4 layers units, 5 layers w

l

• How does the volume of each prism relate to the area of the base

and the number of layers?

e.g., The volume is the number of cubes, which is same as the product

of the base area and the height.

• Why does that make sense?

e.g., Each layer forms another equal group, and you multiply when

there are equal groups.

353

Copyright © 2012 by Nelson Education Ltd. Leaps and Bounds Volume of Prisms: Using a Formula, Pathway 1

2.
Part B

• Make 3 prisms of different heights by stacking pattern blocks.

Use only one type of pattern block for each prism. Sketch your

prisms, or sketch just the base and write the number of layers.

Tower 1: Tower 2: Tower 3:

e.g., e.g., e.g.,

• Describe the volume of each prism, using a triangle block as a unit.

Volume of Tower 1: Volume of Tower 2: Volume of Tower 3:

e.g., 5 green blocks e.g., 1 red block = 3 green e.g., 1 yellow block = 6 green

blocks, so 3 green blocks × 3 blocks, so 6 green blocks × 6

layers = 9 green blocks layers = 36 green blocks

• Estimate the volume of a triangle block, using centimetre cubes.

Explain your thinking.

e.g., I put 4 centimetre cubes in 2 rows of 2. The green block looks

like a bit less than 3 cm3, so my estimate is 2.5 cm3.

• Use your estimate for the volume of the triangle block to estimate

the volume of each prism you built.

Tower 1: Tower 2: Tower 3:

e.g., 2.5 cm3 × 5 green blocks e.g., 2.5 cm3 × 9 green blocks e.g., 2.5 cm3 × 36 green blocks

= 12.5 cm3 = 22.5 cm3 = 90 cm3

• Estimate the area of the base of each of your prisms and multiply

by the height. Then write the result.

Tower 1: Tower 2: Tower 3:

e.g., 2.5 cm2 × 5 cm e.g., 7.5 cm2 × 3 cm e.g., 15 cm2 × 6 cm

= 12.5 cm3 = 22.5 cm3 = 90 cm3

• What do you notice? Why does that make sense?

e.g., I got the same numbers. It makes sense since for rectangular

prisms the volume is the area of the base multiplied by the height.

Volume of Prisms: Using a Formula, Pathway 1 Leaps and Bounds Copyright © 2012 by Nelson Education Ltd.

• Make 3 prisms of different heights by stacking pattern blocks.

Use only one type of pattern block for each prism. Sketch your

prisms, or sketch just the base and write the number of layers.

Tower 1: Tower 2: Tower 3:

e.g., e.g., e.g.,

• Describe the volume of each prism, using a triangle block as a unit.

Volume of Tower 1: Volume of Tower 2: Volume of Tower 3:

e.g., 5 green blocks e.g., 1 red block = 3 green e.g., 1 yellow block = 6 green

blocks, so 3 green blocks × 3 blocks, so 6 green blocks × 6

layers = 9 green blocks layers = 36 green blocks

• Estimate the volume of a triangle block, using centimetre cubes.

Explain your thinking.

e.g., I put 4 centimetre cubes in 2 rows of 2. The green block looks

like a bit less than 3 cm3, so my estimate is 2.5 cm3.

• Use your estimate for the volume of the triangle block to estimate

the volume of each prism you built.

Tower 1: Tower 2: Tower 3:

e.g., 2.5 cm3 × 5 green blocks e.g., 2.5 cm3 × 9 green blocks e.g., 2.5 cm3 × 36 green blocks

= 12.5 cm3 = 22.5 cm3 = 90 cm3

• Estimate the area of the base of each of your prisms and multiply

by the height. Then write the result.

Tower 1: Tower 2: Tower 3:

e.g., 2.5 cm2 × 5 cm e.g., 7.5 cm2 × 3 cm e.g., 15 cm2 × 6 cm

= 12.5 cm3 = 22.5 cm3 = 90 cm3

• What do you notice? Why does that make sense?

e.g., I got the same numbers. It makes sense since for rectangular

prisms the volume is the area of the base multiplied by the height.

Volume of Prisms: Using a Formula, Pathway 1 Leaps and Bounds Copyright © 2012 by Nelson Education Ltd.

3.
Volume of Prisms: Using a Formula Pathway

GUIDED

1

Abby creates chocolate treats in 2 shapes You will need

that are prisms.

• pattern blocks

Abby advertises that the 2 shapes have the • linking cubes

same volume. She knows that because she • a ruler

poured the same number of millilitres of • a calculator

chocolate into each mould.

Re

emember

member

• You can compare the volume of the prisms by looking at the shapes • Volume is the

amount of space a

and sizes in models made from pattern blocks. The area of the

shape takes up.

rhombus is twice the area of the triangle. If you cut the rhombus-

It is measured in

based prism in half and stack the pieces, it will look the same as the cubic units.

triangle-based prism.

• 1 mL of capacity 5

• You can also compare the volume of the prisms by calculating 1 cm3 of volume

the volume of each, using a formula: • Area of a

Volume 5 area of base 3 height parallelogram 5

base 3 height

or V 5 Abase 3 h

• Area of a triangle 5

This is the same as the formula for calculating the volume base 3 height 4 2

of rectangle-based prisms.

Volume of rhombus-based prism: Volume of triangle-based prism:

2.2 cm

2.2 cm

3.0 cm

6.0 cm

2.5 cm

The area of the base is

2.5 3 2.2 5 5.5 cm 2.

For 3 layers, the volume is

5.5 cm2 3 3.0 cm 5 16.5 cm 3. 2.5 cm

The area of the base is

2.5 3 2.2 4 2 5 2.75 cm 2.

For 6 layers, the volume is

2.75 cm2 3 6.0 cm 5 16.5 cm 3.

The volumes are equal.

355

Copyright © 2012 by Nelson Education Ltd. Leaps and Bounds Volume of Prisms: Using a Formula, Pathway 1

GUIDED

1

Abby creates chocolate treats in 2 shapes You will need

that are prisms.

• pattern blocks

Abby advertises that the 2 shapes have the • linking cubes

same volume. She knows that because she • a ruler

poured the same number of millilitres of • a calculator

chocolate into each mould.

Re

emember

member

• You can compare the volume of the prisms by looking at the shapes • Volume is the

amount of space a

and sizes in models made from pattern blocks. The area of the

shape takes up.

rhombus is twice the area of the triangle. If you cut the rhombus-

It is measured in

based prism in half and stack the pieces, it will look the same as the cubic units.

triangle-based prism.

• 1 mL of capacity 5

• You can also compare the volume of the prisms by calculating 1 cm3 of volume

the volume of each, using a formula: • Area of a

Volume 5 area of base 3 height parallelogram 5

base 3 height

or V 5 Abase 3 h

• Area of a triangle 5

This is the same as the formula for calculating the volume base 3 height 4 2

of rectangle-based prisms.

Volume of rhombus-based prism: Volume of triangle-based prism:

2.2 cm

2.2 cm

3.0 cm

6.0 cm

2.5 cm

The area of the base is

2.5 3 2.2 5 5.5 cm 2.

For 3 layers, the volume is

5.5 cm2 3 3.0 cm 5 16.5 cm 3. 2.5 cm

The area of the base is

2.5 3 2.2 4 2 5 2.75 cm 2.

For 6 layers, the volume is

2.75 cm2 3 6.0 cm 5 16.5 cm 3.

The volumes are equal.

355

Copyright © 2012 by Nelson Education Ltd. Leaps and Bounds Volume of Prisms: Using a Formula, Pathway 1

4.
Try These Re

emember

member

. a) Determine the area of the base of a • Assume that the

triangle pattern block. 2.2 cm thickness of each

2 pattern block is 1 cm.

e.g., 2.5 x 2.2 + 2 = 2.75 cm

________________________________ 2.5 cm

• You can figure

b) Determine the area of the base of each pattern block shown. out the height of a

(You can draw lines on the shapes to divide them into triangles.) prism by imagining

identical bases

2.5 cm stacked one on

top of another, or

2.5 cm stacked one behind

another.

2.5 cm

height

e.g., 2.75 × 3 = e.g., 2.75 × 2 = e.g., 2.75 × 6 =

8.25 cm2 5.5 cm2 16.5 cm2

height

c) What is the volume of a stack of 4 trapezoid blocks?

e.g., 8.25 cm2 × 4 cm = 33 cm3

________________________________

d) What is the volume of a stack of 4 blue rhombus blocks?

________________________________

e.g., 5.5 cm2 × 4 cm = 22 cm3

e) What is the volume of a stack of 2 hexagon blocks?

e.g., 16.5 cm2 × 2 cm = 33 cm3

________________________________

2. For each prism below, determine which surface is the base.

Figure out the area of the base, the height, and the volume.

a) b)

12 cm

8 cm

13 cm 3 cm

5 cm

3 cm

area of base: 3 × 8 + 2 = 12 cm2

____________________ area of base: 13 × 12 = 156 cm2

____________________

height of prism: ____________________

5 cm height of prism: ____________________

3 cm

12 × 5 = 60 cm3

volume of prism: ____________________ 156 × 3 = 468 cm3

volume of prism: ____________________

Volume of Prisms: Using a Formula, Pathway 1 Leaps and Bounds Copyright © 2012 by Nelson Education Ltd.

emember

member

. a) Determine the area of the base of a • Assume that the

triangle pattern block. 2.2 cm thickness of each

2 pattern block is 1 cm.

e.g., 2.5 x 2.2 + 2 = 2.75 cm

________________________________ 2.5 cm

• You can figure

b) Determine the area of the base of each pattern block shown. out the height of a

(You can draw lines on the shapes to divide them into triangles.) prism by imagining

identical bases

2.5 cm stacked one on

top of another, or

2.5 cm stacked one behind

another.

2.5 cm

height

e.g., 2.75 × 3 = e.g., 2.75 × 2 = e.g., 2.75 × 6 =

8.25 cm2 5.5 cm2 16.5 cm2

height

c) What is the volume of a stack of 4 trapezoid blocks?

e.g., 8.25 cm2 × 4 cm = 33 cm3

________________________________

d) What is the volume of a stack of 4 blue rhombus blocks?

________________________________

e.g., 5.5 cm2 × 4 cm = 22 cm3

e) What is the volume of a stack of 2 hexagon blocks?

e.g., 16.5 cm2 × 2 cm = 33 cm3

________________________________

2. For each prism below, determine which surface is the base.

Figure out the area of the base, the height, and the volume.

a) b)

12 cm

8 cm

13 cm 3 cm

5 cm

3 cm

area of base: 3 × 8 + 2 = 12 cm2

____________________ area of base: 13 × 12 = 156 cm2

____________________

height of prism: ____________________

5 cm height of prism: ____________________

3 cm

12 × 5 = 60 cm3

volume of prism: ____________________ 156 × 3 = 468 cm3

volume of prism: ____________________

Volume of Prisms: Using a Formula, Pathway 1 Leaps and Bounds Copyright © 2012 by Nelson Education Ltd.

5.
3. a) What is the volume of this prism? 4 cm 2 cm

4 cm

area of base: 4 x 4 + 4 x 2 = 24 cm2 4 cm

2 cm

8 cm

height of prism: __________________

4 cm

24 x 4 = 96 cm3

volume of prism: __________________

b) How does the volume of the prism change if you double the

height to 8 cm?

e.g., The volume doubles. It's 2 of the shapes stacked together.

c) How does the volume change if you double all of the

dimensions?

e.g., The volume is multiplied by 8. (That's 2 x 2 x 2.)

4. This eraser has the shape of a prism with a parallelogram base

that is not a rectangle.

5.5 cm

a) Why is the face with the writing not the base of the prism?

e.g., The base is one of the 2 parallelograms that is not a rectangle,

and the other 4 faces are rectangles.

e.g., about 1 cm

b) Estimate the height of the parallelogram. ___________________

e.g., about 2 cm

Estimate the height of the prism. ___________________

5. Suppose Marla made a wooden sculpture in the shape of a

parallelogram-based prism. It has a volume of about 150 cm3.

Sketch 2 possible drawings for the base and label the

dimensions. Determine the height of each prism.

Area of base: 10 × 5 = 50 cm2 Area of base: 10 × 3 = 30 cm2

Prism height: 3 cm Prism height: 5 cm

357

Copyright © 2012 by Nelson Education Ltd. Leaps and Bounds Volume of Prisms: Using a Formula, Pathway 1

4 cm

area of base: 4 x 4 + 4 x 2 = 24 cm2 4 cm

2 cm

8 cm

height of prism: __________________

4 cm

24 x 4 = 96 cm3

volume of prism: __________________

b) How does the volume of the prism change if you double the

height to 8 cm?

e.g., The volume doubles. It's 2 of the shapes stacked together.

c) How does the volume change if you double all of the

dimensions?

e.g., The volume is multiplied by 8. (That's 2 x 2 x 2.)

4. This eraser has the shape of a prism with a parallelogram base

that is not a rectangle.

5.5 cm

a) Why is the face with the writing not the base of the prism?

e.g., The base is one of the 2 parallelograms that is not a rectangle,

and the other 4 faces are rectangles.

e.g., about 1 cm

b) Estimate the height of the parallelogram. ___________________

e.g., about 2 cm

Estimate the height of the prism. ___________________

5. Suppose Marla made a wooden sculpture in the shape of a

parallelogram-based prism. It has a volume of about 150 cm3.

Sketch 2 possible drawings for the base and label the

dimensions. Determine the height of each prism.

Area of base: 10 × 5 = 50 cm2 Area of base: 10 × 3 = 30 cm2

Prism height: 3 cm Prism height: 5 cm

357

Copyright © 2012 by Nelson Education Ltd. Leaps and Bounds Volume of Prisms: Using a Formula, Pathway 1

6.
6. Some of the measurements of a prism are in metres and some

are in centimetres. Why would you convert the measurements to

the same units?

e.g., If you multiplied the numbers without converting, the result would

not be meaningful; It wouldn't be square centimetres or square metres.

7. What is the volume of concrete needed to build these

steps?

1.3 m

height

e.g., bottom layer: 1.3 × 0.2 × 1 = 0.26 m2

of each

2nd layer: 1.3 × 0.2 × 0.75 = 0.2 m3 step:

3rd layer: 1.3 × 0.2 × 0.5 = 0.13 m3 20 cm

top layer: 1.3 × 0.2 × 0.25 = 0.07 m3

Total volume: 0.66 m3

depth of each step: 25 cm

8. These earrings are shaped like triangle-based prisms. What is the

volume of the 2 earrings?

e.g., The area of each base is 1 × 0.9 ÷ 2 = 0.45 cm2.

The height is 4 cm. The volume of 1 prism is 0.45 × 4 = 1.8 cm3.

The total volume is 2 × 1.8 = 3.6 cm3.

4 cm

0.9 cm

1 cm

9. Two other prisms have the same volume as the one shown

below. None of the prisms have a rectangular base. Sketch

2 possible bases for the other prisms and label the dimensions.

Determine the height of each prism. FYI

Learning how to

area of base: 6 × 5 ÷ 2 = 15 cm2

___________________ calculate volumes of

prisms will be useful

8 cm

___________________ for learning how to

5 cm height of prism:

8 cm calculate volumes of

cylinders later on.

6 cm 15 × 8 = 120 cm3

volume of prism: ___________________

Area of base: 2 × 15 ÷ 2 = 15 cm2 Area of base: 5 × 12 ÷ 2 = 30 cm2

Prism height: 8 cm Prism height: 4 cm

Volume of Prisms: Using a Formula, Pathway 1 Leaps and Bounds Copyright © 2012 by Nelson Education Ltd.

are in centimetres. Why would you convert the measurements to

the same units?

e.g., If you multiplied the numbers without converting, the result would

not be meaningful; It wouldn't be square centimetres or square metres.

7. What is the volume of concrete needed to build these

steps?

1.3 m

height

e.g., bottom layer: 1.3 × 0.2 × 1 = 0.26 m2

of each

2nd layer: 1.3 × 0.2 × 0.75 = 0.2 m3 step:

3rd layer: 1.3 × 0.2 × 0.5 = 0.13 m3 20 cm

top layer: 1.3 × 0.2 × 0.25 = 0.07 m3

Total volume: 0.66 m3

depth of each step: 25 cm

8. These earrings are shaped like triangle-based prisms. What is the

volume of the 2 earrings?

e.g., The area of each base is 1 × 0.9 ÷ 2 = 0.45 cm2.

The height is 4 cm. The volume of 1 prism is 0.45 × 4 = 1.8 cm3.

The total volume is 2 × 1.8 = 3.6 cm3.

4 cm

0.9 cm

1 cm

9. Two other prisms have the same volume as the one shown

below. None of the prisms have a rectangular base. Sketch

2 possible bases for the other prisms and label the dimensions.

Determine the height of each prism. FYI

Learning how to

area of base: 6 × 5 ÷ 2 = 15 cm2

___________________ calculate volumes of

prisms will be useful

8 cm

___________________ for learning how to

5 cm height of prism:

8 cm calculate volumes of

cylinders later on.

6 cm 15 × 8 = 120 cm3

volume of prism: ___________________

Area of base: 2 × 15 ÷ 2 = 15 cm2 Area of base: 5 × 12 ÷ 2 = 30 cm2

Prism height: 8 cm Prism height: 4 cm

Volume of Prisms: Using a Formula, Pathway 1 Leaps and Bounds Copyright © 2012 by Nelson Education Ltd.

7.
Surface Area of Prisms Pathway

OPEN-ENDED

2

Aviva has 200 cm2 of fancy paper You will need

that she will use to cover a prism. I don’t want to

waste any paper • prisms from a set of

• Think of 2 different-shaped prisms that geometric solids

would each use all or most of Aviva’s paper overlapping it, so

• connecting faces

to cover the faces. Sketch the prisms and I’ll cut out pieces • 1 cm Grid Paper

label the dimensions you think are best for and glue them on. (BLM 12, optional)

each prism. • a ruler

• a calculator

(You might look at models and make a

sketch of each prism.

Or you might use connecting faces to make Re

emember

member

a model of each prism. • Area is measured in

Or you might sketch a net of each prism on square units.

grid paper.) • The base of a prism

is one of the 2 shapes

in the prism that may

not be a rectangle.

There are always

2 identical bases—

the top and bottom.

• The words base and

height can mean

Prism 1 sketch and dimensions: 2 different things

e.g., I think a square-based prism might work. in a prism.

The base can be a

polygon at the base

of the prism (B), or a

side length (b) of that

polygon.

The height can be

the full height of the

prism, or the height of

a base of the prism.

h

Prism 2 sketch and dimensions:

B

e.g., I think a triangle-based prism might work. h b

• A net is a 2-D picture

of a 3-D object that

is folded down.

359

Copyright © 2012 by Nelson Education Ltd. Leaps and Bounds Surface Area of Prisms, Pathway 2

OPEN-ENDED

2

Aviva has 200 cm2 of fancy paper You will need

that she will use to cover a prism. I don’t want to

waste any paper • prisms from a set of

• Think of 2 different-shaped prisms that geometric solids

would each use all or most of Aviva’s paper overlapping it, so

• connecting faces

to cover the faces. Sketch the prisms and I’ll cut out pieces • 1 cm Grid Paper

label the dimensions you think are best for and glue them on. (BLM 12, optional)

each prism. • a ruler

• a calculator

(You might look at models and make a

sketch of each prism.

Or you might use connecting faces to make Re

emember

member

a model of each prism. • Area is measured in

Or you might sketch a net of each prism on square units.

grid paper.) • The base of a prism

is one of the 2 shapes

in the prism that may

not be a rectangle.

There are always

2 identical bases—

the top and bottom.

• The words base and

height can mean

Prism 1 sketch and dimensions: 2 different things

e.g., I think a square-based prism might work. in a prism.

The base can be a

polygon at the base

of the prism (B), or a

side length (b) of that

polygon.

The height can be

the full height of the

prism, or the height of

a base of the prism.

h

Prism 2 sketch and dimensions:

B

e.g., I think a triangle-based prism might work. h b

• A net is a 2-D picture

of a 3-D object that

is folded down.

359

Copyright © 2012 by Nelson Education Ltd. Leaps and Bounds Surface Area of Prisms, Pathway 2

8.
• Determine the surface area of each prism.

Use sketches to help you. surface area

the total area of all of

the faces of a shape

Prism 1 surface area:

e.g., The rectangle base is 5 × 5, so the top and

bottom areas together are 2 × 25 = 50 cm2.

Each side face has area 35 cm2. Since there are

4 of them, the area of all of the rectangles is

35 × 4 = 140 cm2.

So the total area is 50 + 140 = 190 cm2.

That's close to 200 cm2.

Prism 2 surface area:

e.g., The triangle base has an area of 24, so the top

and bottom together have an area of 48 cm2.

The side rectangles have areas of 36, 48, and 60, so

the total is 192 cm2. That's even closer to 200 cm2.

• Why did some pairs of faces have the same area?

I made this model using

e.g., The top and bottom always have the same area since in prisms 6 connecting faces

the top and bottom are identical. Sometimes if the base had other

equal sides, there would be other equal rectangular faces.

that are the same.

Surface Area of Prisms, Pathway 2 Leaps and Bounds Copyright © 2012 by Nelson Education Ltd.

Use sketches to help you. surface area

the total area of all of

the faces of a shape

Prism 1 surface area:

e.g., The rectangle base is 5 × 5, so the top and

bottom areas together are 2 × 25 = 50 cm2.

Each side face has area 35 cm2. Since there are

4 of them, the area of all of the rectangles is

35 × 4 = 140 cm2.

So the total area is 50 + 140 = 190 cm2.

That's close to 200 cm2.

Prism 2 surface area:

e.g., The triangle base has an area of 24, so the top

and bottom together have an area of 48 cm2.

The side rectangles have areas of 36, 48, and 60, so

the total is 192 cm2. That's even closer to 200 cm2.

• Why did some pairs of faces have the same area?

I made this model using

e.g., The top and bottom always have the same area since in prisms 6 connecting faces

the top and bottom are identical. Sometimes if the base had other

equal sides, there would be other equal rectangular faces.

that are the same.

Surface Area of Prisms, Pathway 2 Leaps and Bounds Copyright © 2012 by Nelson Education Ltd.

9.
Surface Area of Prisms Pathway

GUIDED

2

Scientists know that the greater the You will need

area of the leaves of a plant, the

more sunlight can be absorbed for • prisms from a set of

growth. The surface area of plants geometric solids

• a calculator

can be complicated to figure out,

• a tape measure or a

but the surface area of shapes like

metre stick

prisms is easier to calculate.

Every prism has 2 bases with the

same area. It also has a number

surface area

of rectangular faces that might or

the total area of all of

might not have the same area. the faces of a shape

Re

emember

member

• Area is measured in

square units.

• To calculate the surface area of a 3-D shape, you add all of the areas.

• A net is a 2-D picture

For example, look at the triangle-based prism and its net, below. of a 3-D object that

is folded down.

4 6 = 24 cm2 3 6 = 18 cm2

5 cm

4 cm

6 cm

3 cm

5 6 = 30 cm2

3 4 2 2 = 12 cm 2

You can calculate the surface area (SA) of the triangle-based prism

like this:

SA 5 area of top base 1 area of bottom base 1 area of 3 rectangles

5 (4 3 3 4 2) 1 (4 3 3 1 2) 1 (6 3 4) 1 (6 3 3) 1 (6 3 5)

5 6 1 6 1 24 1 18 1 30

5 84 cm 2

For any prism: Re

emember

member

SA 5 2 3 area of base 1 area of all side rectangles • Area of a

The number of rectangles is always the same parallelogram 5

as the number of sides of the base. base 3 height

• Area of a triangle 5

• How many areas do you add to determine base 3 height 4 2

the area of a hexagon-based prism?

(oral) e.g., 8 areas for 8 faces

361

Copyright © 2012 by Nelson Education Ltd. Leaps and Bounds Surface Area of Prisms, Pathway 2

GUIDED

2

Scientists know that the greater the You will need

area of the leaves of a plant, the

more sunlight can be absorbed for • prisms from a set of

growth. The surface area of plants geometric solids

• a calculator

can be complicated to figure out,

• a tape measure or a

but the surface area of shapes like

metre stick

prisms is easier to calculate.

Every prism has 2 bases with the

same area. It also has a number

surface area

of rectangular faces that might or

the total area of all of

might not have the same area. the faces of a shape

Re

emember

member

• Area is measured in

square units.

• To calculate the surface area of a 3-D shape, you add all of the areas.

• A net is a 2-D picture

For example, look at the triangle-based prism and its net, below. of a 3-D object that

is folded down.

4 6 = 24 cm2 3 6 = 18 cm2

5 cm

4 cm

6 cm

3 cm

5 6 = 30 cm2

3 4 2 2 = 12 cm 2

You can calculate the surface area (SA) of the triangle-based prism

like this:

SA 5 area of top base 1 area of bottom base 1 area of 3 rectangles

5 (4 3 3 4 2) 1 (4 3 3 1 2) 1 (6 3 4) 1 (6 3 3) 1 (6 3 5)

5 6 1 6 1 24 1 18 1 30

5 84 cm 2

For any prism: Re

emember

member

SA 5 2 3 area of base 1 area of all side rectangles • Area of a

The number of rectangles is always the same parallelogram 5

as the number of sides of the base. base 3 height

• Area of a triangle 5

• How many areas do you add to determine base 3 height 4 2

the area of a hexagon-based prism?

(oral) e.g., 8 areas for 8 faces

361

Copyright © 2012 by Nelson Education Ltd. Leaps and Bounds Surface Area of Prisms, Pathway 2

10.
Try These

Identify a base of each prism.

Figure out the area of the base. Re

emember

member

• The base of a

a) rectangle-based prism c) isosceles triangle-based prism is one of the

prism 2 shapes in the

prism that may not

10 cm be rectangles.

9 cm

8 cm There are always

2 identical bases—

the top and bottom.

3 cm

7 cm • The words base and

7 cm height can mean

12 cm

2 different things

area of base: area of base: in a prism.

The base can be a

e.g., 7 × 3 = 21 cm2

____________________ 12 × 8 ÷ 2 = 48 cm2

____________________

polygon at the base

of the prism (B), or a

b) parallelogram-based prism d) regular hexagon-based side length (b) of that

prism polygon.

The height can be

4 cm

the full height of the

5 cm prism, or the height

8 cm of a base of the

6 cm prism.

5.2 cm

h

4 cm

6 cm B

h b

area of base: area of base:

____________________

6 × 3 = 18 cm2 (6 × 5.2 ÷ 2) × 6 = 93.6 cm2

____________________

2. Calculate the areas of the faces that you did not identify as bases

in Question 1. (These faces should all be rectangles.)

a) c)

e.g., 2 faces: 3 × 9 = 27 cm2 2 faces: 7 × 10 = 70 cm2

2 faces: 7 × 9 = 63 cm2 1 face: 12 × 7 = 84 cm2

b) d)

2 faces: 6 × 8 = 48 cm2 6 faces: 6 × 4 = 24 cm2

2 faces: 5 × 8 = 40 cm2

Surface Area of Prisms, Pathway 2 Leaps and Bounds Copyright © 2012 by Nelson Education Ltd.

Identify a base of each prism.

Figure out the area of the base. Re

emember

member

• The base of a

a) rectangle-based prism c) isosceles triangle-based prism is one of the

prism 2 shapes in the

prism that may not

10 cm be rectangles.

9 cm

8 cm There are always

2 identical bases—

the top and bottom.

3 cm

7 cm • The words base and

7 cm height can mean

12 cm

2 different things

area of base: area of base: in a prism.

The base can be a

e.g., 7 × 3 = 21 cm2

____________________ 12 × 8 ÷ 2 = 48 cm2

____________________

polygon at the base

of the prism (B), or a

b) parallelogram-based prism d) regular hexagon-based side length (b) of that

prism polygon.

The height can be

4 cm

the full height of the

5 cm prism, or the height

8 cm of a base of the

6 cm prism.

5.2 cm

h

4 cm

6 cm B

h b

area of base: area of base:

____________________

6 × 3 = 18 cm2 (6 × 5.2 ÷ 2) × 6 = 93.6 cm2

____________________

2. Calculate the areas of the faces that you did not identify as bases

in Question 1. (These faces should all be rectangles.)

a) c)

e.g., 2 faces: 3 × 9 = 27 cm2 2 faces: 7 × 10 = 70 cm2

2 faces: 7 × 9 = 63 cm2 1 face: 12 × 7 = 84 cm2

b) d)

2 faces: 6 × 8 = 48 cm2 6 faces: 6 × 4 = 24 cm2

2 faces: 5 × 8 = 40 cm2

Surface Area of Prisms, Pathway 2 Leaps and Bounds Copyright © 2012 by Nelson Education Ltd.

11.
3. Calculate the surface area for each prism in Question 1.

a) c)

(2 × 21) + (2 × 27) + (2 × 63) (2 × 48) + (2 × 70) + 84

= 222 cm2 = 320 cm2

b) d)

(2 × 18) + (2 × 48) + (2 × 40) (2 × 93.6) + (6 × 24)

= 331.2 cm2

= 212 cm2

4. What is the surface area of each rectangular box?

Do not plan for any overlap.

a) b)

4 cm 6 cm

3 cm

7 cm

12 cm

10 cm

2 × (7 × 4) + 2 × (7 × 3) + 2 × (3 × 4) 2 × (10 × 6) + 2 × (10 × 12) + 2 × (12 × 6)

= 122 cm2 = 504 cm2

5. Suppose you need to know the surface area of a rectangular

classroom to paint it.

a) Estimate the surface area of the walls and ceiling of the

classroom. Do not include the floor. Show your thinking.

e.g., The room is about 8 m × 7 m, so the ceiling area is 56 m2.

The room is about 3 m high, so the walls' area is

(2 × 24) + (2 × 21) = 90 m2. The total surface area without the

floor is 146 m2.

b) If a 4 L can of paint covers 36 m2, about how many cans of

paint would you need?

146 ÷ 36 is about 4 cans of paint

363

Copyright © 2012 by Nelson Education Ltd. Leaps and Bounds Surface Area of Prisms, Pathway 2

a) c)

(2 × 21) + (2 × 27) + (2 × 63) (2 × 48) + (2 × 70) + 84

= 222 cm2 = 320 cm2

b) d)

(2 × 18) + (2 × 48) + (2 × 40) (2 × 93.6) + (6 × 24)

= 331.2 cm2

= 212 cm2

4. What is the surface area of each rectangular box?

Do not plan for any overlap.

a) b)

4 cm 6 cm

3 cm

7 cm

12 cm

10 cm

2 × (7 × 4) + 2 × (7 × 3) + 2 × (3 × 4) 2 × (10 × 6) + 2 × (10 × 12) + 2 × (12 × 6)

= 122 cm2 = 504 cm2

5. Suppose you need to know the surface area of a rectangular

classroom to paint it.

a) Estimate the surface area of the walls and ceiling of the

classroom. Do not include the floor. Show your thinking.

e.g., The room is about 8 m × 7 m, so the ceiling area is 56 m2.

The room is about 3 m high, so the walls' area is

(2 × 24) + (2 × 21) = 90 m2. The total surface area without the

floor is 146 m2.

b) If a 4 L can of paint covers 36 m2, about how many cans of

paint would you need?

146 ÷ 36 is about 4 cans of paint

363

Copyright © 2012 by Nelson Education Ltd. Leaps and Bounds Surface Area of Prisms, Pathway 2

12.
6. Suppose you have one of the prisms below. You can calculate the

total surface area by doubling one area and tripling another, and

then adding them. What might the shape look like? Circle one.

7. To determine the surface area of some shapes, Aidan performed

the following calculations. What might each shape have been?

Use a name or a sketch.

a) 6 3 (7 3 7) c) 2 3 (5 3 12 4 2) 1 5 3 8 1 12 3 8 1

13 3 8

e.g., a cube, 7 by 7 by 7 e.g., a triangle-based prism with height

8, and base with height 12 and side

length 5

b) 2 3 (3 3 5) 1 2 3 (3 3 7) 1 2 3 (3 3 5) d) 2 3 (6 3 5.2 4 2) 1 (6 3 6) 3 5

e.g., a rectangular prism, 3 by 5 by 7 e.g., a regular hexagon-based prism

with base side of 6 and height of 5

8. You can use 20 linking cubes to make a 1-by-1-by-20 prism or

a 2-by-2-by-5 prism. They have the same volume.

Do they have the same surface area? How do you know?

No. e.g., One has a surface area of 82 square units, but the other

has a surface area of 48 square units.

9. A rectangular box has a surface area of 52 cm2.

What might the length, width, and height be?

e.g.,

FYI

The surface area of an

animal or a human is

an important factor in

keeping them neither

too hot nor too cold.

Surface Area of Prisms, Pathway 2 Leaps and Bounds Copyright © 2012 by Nelson Education Ltd.

total surface area by doubling one area and tripling another, and

then adding them. What might the shape look like? Circle one.

7. To determine the surface area of some shapes, Aidan performed

the following calculations. What might each shape have been?

Use a name or a sketch.

a) 6 3 (7 3 7) c) 2 3 (5 3 12 4 2) 1 5 3 8 1 12 3 8 1

13 3 8

e.g., a cube, 7 by 7 by 7 e.g., a triangle-based prism with height

8, and base with height 12 and side

length 5

b) 2 3 (3 3 5) 1 2 3 (3 3 7) 1 2 3 (3 3 5) d) 2 3 (6 3 5.2 4 2) 1 (6 3 6) 3 5

e.g., a rectangular prism, 3 by 5 by 7 e.g., a regular hexagon-based prism

with base side of 6 and height of 5

8. You can use 20 linking cubes to make a 1-by-1-by-20 prism or

a 2-by-2-by-5 prism. They have the same volume.

Do they have the same surface area? How do you know?

No. e.g., One has a surface area of 82 square units, but the other

has a surface area of 48 square units.

9. A rectangular box has a surface area of 52 cm2.

What might the length, width, and height be?

e.g.,

FYI

The surface area of an

animal or a human is

an important factor in

keeping them neither

too hot nor too cold.

Surface Area of Prisms, Pathway 2 Leaps and Bounds Copyright © 2012 by Nelson Education Ltd.

13.
Volume of Rectangular Prisms Pathway

OPEN-ENDED

3

Rectangular prisms are very common shapes. You will need

• 50 linking cubes

• a calculator

• 3 classroom objects

in the shape of

rectangular prisms

• a ruler

Part A

• Use 50 linking cubes to build 3 rectangular prisms with different

dimensions. Use all or most of the cubes to build all 3 prisms.

• Record the number of cubes in the length, width, and height of each

prism, and the volume of the prism. volume

the amount of space

Prism 1: Prism 2: Prism 3: that a 3-D shape

e.g., 4 by 3 by 2, e.g., 1 by 1 by 10, takes up, measured

e.g., 4 by 2 by 2, in cubic units

volume is 16 cubic volume is 24 cubic volume is 10 cubic

units units units

• Repeat the task above for 3 more prisms.

Prism 4: Prism 5: Prism 6:

e.g., 2 by 2 by 2, e.g., 3 by 3 by 3, e.g., 2 by 5 by 1,

volume is 8 cubic volume is 27 cubic volume is 10 cubic

units units units

Re

emember

member

• Typical units of

volume are cubic

centimetres (cm3) and

• How can you predict the volume of the prism if you know the area of cubic metres (m3).

the base and the height? Tell why this makes sense. • Units for the area of

the base are likely

e.g., You can multiply the length and width to get the area of the base, square centimetres

and then multiply by the height to get the volume. Volume is a 3-D (cm2) or square

measurement so you use 3 dimensions. metres (m2).

• Area of a rectangle

5 length 3 width

365

Copyright © 2012 by Nelson Education Ltd. Leaps and Bounds Volume of Rectangular Prisms, Pathway 3

OPEN-ENDED

3

Rectangular prisms are very common shapes. You will need

• 50 linking cubes

• a calculator

• 3 classroom objects

in the shape of

rectangular prisms

• a ruler

Part A

• Use 50 linking cubes to build 3 rectangular prisms with different

dimensions. Use all or most of the cubes to build all 3 prisms.

• Record the number of cubes in the length, width, and height of each

prism, and the volume of the prism. volume

the amount of space

Prism 1: Prism 2: Prism 3: that a 3-D shape

e.g., 4 by 3 by 2, e.g., 1 by 1 by 10, takes up, measured

e.g., 4 by 2 by 2, in cubic units

volume is 16 cubic volume is 24 cubic volume is 10 cubic

units units units

• Repeat the task above for 3 more prisms.

Prism 4: Prism 5: Prism 6:

e.g., 2 by 2 by 2, e.g., 3 by 3 by 3, e.g., 2 by 5 by 1,

volume is 8 cubic volume is 27 cubic volume is 10 cubic

units units units

Re

emember

member

• Typical units of

volume are cubic

centimetres (cm3) and

• How can you predict the volume of the prism if you know the area of cubic metres (m3).

the base and the height? Tell why this makes sense. • Units for the area of

the base are likely

e.g., You can multiply the length and width to get the area of the base, square centimetres

and then multiply by the height to get the volume. Volume is a 3-D (cm2) or square

measurement so you use 3 dimensions. metres (m2).

• Area of a rectangle

5 length 3 width

365

Copyright © 2012 by Nelson Education Ltd. Leaps and Bounds Volume of Rectangular Prisms, Pathway 3

14.
Part B

Re

emember

member

• Find 3 objects in the classroom that are not too big and look like

• The volume of

rectangular prisms. (Or, imagine a rectangular box that your object

a linking cube is

would fit inside with very little extra space.) about 8 cm3.

• Measure and record the length, width, and height in centimetres.

• Predict the volume in cubic centimetres.

• Use cubes to check.

Object 1:

e.g., The stapler is about 18 cm × 4 cm × 5 cm high.

I predict 360 cm3 since 18 × 4 × 5 = 360.

My linking cube prism was 9 cubes × 2 cubes × 3 cubes = 54 cubes, and 8 × 54 = 432 cm3.

I knew it was a bit too high, so 360 seemed okay.

Object 2:

e.g., The whiteboard eraser is about 11 cm × 5 cm × 3 cm high.

I predict 165 cm3.

I built a 5 × 3 × 2 linking cube prism. It took 30 cubes and I knew it was bigger than the eraser.

30 × 8 = 240 cm3 and that is bigger, so I think 165 is good.

Object 3:

e.g., The stack of CDs is about 12 cm × 13 cm × 14 cm high. I predict 2184 cm3.

I didn't check with linking cubes since I'd need too many, but I did make a 6 × 7 layer and

realized there would be about 6 of them or a bit more; 6 × 7 × 6 = 252 and 8 × 252 = 2016, so

2184 seemed okay.

Volume of Rectangular Prisms, Pathway 3 Leaps and Bounds Copyright © 2012 by Nelson Education Ltd.

Re

emember

member

• Find 3 objects in the classroom that are not too big and look like

• The volume of

rectangular prisms. (Or, imagine a rectangular box that your object

a linking cube is

would fit inside with very little extra space.) about 8 cm3.

• Measure and record the length, width, and height in centimetres.

• Predict the volume in cubic centimetres.

• Use cubes to check.

Object 1:

e.g., The stapler is about 18 cm × 4 cm × 5 cm high.

I predict 360 cm3 since 18 × 4 × 5 = 360.

My linking cube prism was 9 cubes × 2 cubes × 3 cubes = 54 cubes, and 8 × 54 = 432 cm3.

I knew it was a bit too high, so 360 seemed okay.

Object 2:

e.g., The whiteboard eraser is about 11 cm × 5 cm × 3 cm high.

I predict 165 cm3.

I built a 5 × 3 × 2 linking cube prism. It took 30 cubes and I knew it was bigger than the eraser.

30 × 8 = 240 cm3 and that is bigger, so I think 165 is good.

Object 3:

e.g., The stack of CDs is about 12 cm × 13 cm × 14 cm high. I predict 2184 cm3.

I didn't check with linking cubes since I'd need too many, but I did make a 6 × 7 layer and

realized there would be about 6 of them or a bit more; 6 × 7 × 6 = 252 and 8 × 252 = 2016, so

2184 seemed okay.

Volume of Rectangular Prisms, Pathway 3 Leaps and Bounds Copyright © 2012 by Nelson Education Ltd.

15.
Volume of Rectangular Prisms Pathway

GUIDED

3

The volume of a stack of books depends on the size of the pages and You will need

the height of the stack.

• centimetre cubes or

linking cubes

• a calculator

• a ruler

volume

the amount of space

that a 3-D shape

takes up, measured

• You can use centimetre cubes to model each stack. in cubic units

The short stack is made of

3 layers of 12 3 18 5 216 cubes.

Since this stack requires

3 3 216 5 648 centimetre cubes, Re

emember

member

the volume is 648 cm3. • Typical units of

volume are cubic

centimetres (cm3)

The tall stack with the same page size and cubic metres (m3).

can be modelled with • Units for the area of

6 layers of 216 cubes. the base are likely

Its volume is 6 3 216 5 1296 cm3. square centimetres

(cm2) or square

metres (m2).

The tall stack with a larger page size

can be modelled with

6 layers of 12 3 22 5 264 cubes.

Its volume is 6 3 264 5 1584 cm3.

The volume of a rectangular prism is calculated by multiplying

the area of the base by the height.

Volume 5 area of base 3 height or

V 5 Abase 3 h

367

Copyright © 2012 by Nelson Education Ltd. Leaps and Bounds Volume of Rectangular Prisms, Pathway 3

GUIDED

3

The volume of a stack of books depends on the size of the pages and You will need

the height of the stack.

• centimetre cubes or

linking cubes

• a calculator

• a ruler

volume

the amount of space

that a 3-D shape

takes up, measured

• You can use centimetre cubes to model each stack. in cubic units

The short stack is made of

3 layers of 12 3 18 5 216 cubes.

Since this stack requires

3 3 216 5 648 centimetre cubes, Re

emember

member

the volume is 648 cm3. • Typical units of

volume are cubic

centimetres (cm3)

The tall stack with the same page size and cubic metres (m3).

can be modelled with • Units for the area of

6 layers of 216 cubes. the base are likely

Its volume is 6 3 216 5 1296 cm3. square centimetres

(cm2) or square

metres (m2).

The tall stack with a larger page size

can be modelled with

6 layers of 12 3 22 5 264 cubes.

Its volume is 6 3 264 5 1584 cm3.

The volume of a rectangular prism is calculated by multiplying

the area of the base by the height.

Volume 5 area of base 3 height or

V 5 Abase 3 h

367

Copyright © 2012 by Nelson Education Ltd. Leaps and Bounds Volume of Rectangular Prisms, Pathway 3

16.
Try These

. Each prism below is made of centimetre cubes. What is the

area of the base of the prism? What is the height of the prism

(the number of layers)? What is the volume?

a) c)

area of base: 6 cm2

____________________

area of base: 24 cm2

____________________

height of prism: ____________________

4 cm

height of prism: 3 cm

____________________

6 × 4 = 24 cm3

volume of prism: ____________________

24 × 3 = 72 cm3

volume of prism: ____________________

b) d)

area of base: 15 cm2

____________________

height of prism: ____________________

4 cm

area of base: 24 cm2

____________________

15 × 4 = 60 cm3

volume of prism: ____________________

height of prism: ____________________

5 cm

24 × 5 = 120 cm3

volume of prism: ____________________

Volume of Rectangular Prisms, Pathway 3 Leaps and Bounds Copyright © 2012 by Nelson Education Ltd.

. Each prism below is made of centimetre cubes. What is the

area of the base of the prism? What is the height of the prism

(the number of layers)? What is the volume?

a) c)

area of base: 6 cm2

____________________

area of base: 24 cm2

____________________

height of prism: ____________________

4 cm

height of prism: 3 cm

____________________

6 × 4 = 24 cm3

volume of prism: ____________________

24 × 3 = 72 cm3

volume of prism: ____________________

b) d)

area of base: 15 cm2

____________________

height of prism: ____________________

4 cm

area of base: 24 cm2

____________________

15 × 4 = 60 cm3

volume of prism: ____________________

height of prism: ____________________

5 cm

24 × 5 = 120 cm3

volume of prism: ____________________

Volume of Rectangular Prisms, Pathway 3 Leaps and Bounds Copyright © 2012 by Nelson Education Ltd.

17.
2. a) Use centimetre cubes to build a prism with a 3-by-4 base

and a height of 2. Calculate the volume and explain your

reasoning.

e.g., The volume is 24 cm3 since the area of the base is 12 cm2 and

the height is 2 cm.

b) Turn the prism so the base is the 4-by-2 face.

What is the height? What is the volume?

e.g., The height is 3 cm and the volume is still 24 cm3.

c) Turn the prism again so the base is the 2-by-3 face.

What is the height? What is the volume?

e.g., The height is 4 cm and the volume is still 24 cm3.

3. A gift box has the given volume. What might its length, width, and

height be?

a) 500 cm3 b) 1350 cm3

e.g., 10 cm long, 10 cm e.g., 15 cm long, 9 cm

wide, and 5 cm high high, and 10 cm wide

4. A TV remote control in the shape of a rectangular prism has a

volume of 132 cm3. What might the height, width, and length be?

Show 2 possible solutions.

e.g., 22 cm long, 2 cm high, 3 cm wide;

11 cm long, 4 cm high, 3 cm wide

5. A wedding cake is made up of 3 tiers. Each tier is a

8 cm

square-based prism as shown and is 8 cm high.

What is the total volume of the cake? 15 cm 8 cm

bottom tier: 36 × 36 × 8 = 10 368 cm3 8 cm

middle tier: 25 × 25 × 8 = 5000 cm3 25 cm

top tier: 15 × 15 × 8 = 1800 cm3

Total volume = 17 168 cm3

36 cm

369

Copyright © 2012 by Nelson Education Ltd. Leaps and Bounds Volume of Rectangular Prisms, Pathway 3

and a height of 2. Calculate the volume and explain your

reasoning.

e.g., The volume is 24 cm3 since the area of the base is 12 cm2 and

the height is 2 cm.

b) Turn the prism so the base is the 4-by-2 face.

What is the height? What is the volume?

e.g., The height is 3 cm and the volume is still 24 cm3.

c) Turn the prism again so the base is the 2-by-3 face.

What is the height? What is the volume?

e.g., The height is 4 cm and the volume is still 24 cm3.

3. A gift box has the given volume. What might its length, width, and

height be?

a) 500 cm3 b) 1350 cm3

e.g., 10 cm long, 10 cm e.g., 15 cm long, 9 cm

wide, and 5 cm high high, and 10 cm wide

4. A TV remote control in the shape of a rectangular prism has a

volume of 132 cm3. What might the height, width, and length be?

Show 2 possible solutions.

e.g., 22 cm long, 2 cm high, 3 cm wide;

11 cm long, 4 cm high, 3 cm wide

5. A wedding cake is made up of 3 tiers. Each tier is a

8 cm

square-based prism as shown and is 8 cm high.

What is the total volume of the cake? 15 cm 8 cm

bottom tier: 36 × 36 × 8 = 10 368 cm3 8 cm

middle tier: 25 × 25 × 8 = 5000 cm3 25 cm

top tier: 15 × 15 × 8 = 1800 cm3

Total volume = 17 168 cm3

36 cm

369

Copyright © 2012 by Nelson Education Ltd. Leaps and Bounds Volume of Rectangular Prisms, Pathway 3

18.
6. a) What is the volume of this cube?

10 × 10 × 10 = 1000 cm3

____________________

10 cm

b) What is the volume of the

remaining part of the cube after 10 cm

the smaller cube is taken out? 10 cm

10 cm

10 cm 3 cm

3 cm

10 cm 3 cm

10 × 10 × 10 - 3 × 3 × 3 = 973 cm3

7. Explain why it makes sense that this prism has a volume

of 42 cm3. 3.5 cm

e.g., The area of base is 4 × 3 = 12 cm2. If you multiply by 3, you get

3.0 cm

36 cm3 and if you multiply by 4, you get 48 cm3, so I went halfway 4.0 cm

between.

8. Using centimetre cubes, can you build more prisms with a volume

of 12 cm3 or with a volume of 14 cm3? Explain your thinking.

e.g., There are more prisms for 12 since it could be 1 × 1 × 12,

or 1 × 2 × 6, or 1 × 3 × 4, but for 14 it can only be 1 × 1 × 14 or 1 × 2 × 7.

9. The formula for the volume of a rectangular prism can be written

as V 5 Abase 3 h.

Why is the formula below another formula for the volume of

a rectangular prism?

V 5 l 3 w 3 h, where l is the length of the base, w is the width of

the base, and h is the height of the prism

FYI

Learning how to

e.g., The area of the base is l × w, so you can just say that instead calculate volumes of

rectangular prisms

of Abase before you multiply by the height. will help you learn

formulas for other

prisms, too.

Volume of Rectangular Prisms, Pathway 3 Leaps and Bounds Copyright © 2012 by Nelson Education Ltd.

10 × 10 × 10 = 1000 cm3

____________________

10 cm

b) What is the volume of the

remaining part of the cube after 10 cm

the smaller cube is taken out? 10 cm

10 cm

10 cm 3 cm

3 cm

10 cm 3 cm

10 × 10 × 10 - 3 × 3 × 3 = 973 cm3

7. Explain why it makes sense that this prism has a volume

of 42 cm3. 3.5 cm

e.g., The area of base is 4 × 3 = 12 cm2. If you multiply by 3, you get

3.0 cm

36 cm3 and if you multiply by 4, you get 48 cm3, so I went halfway 4.0 cm

between.

8. Using centimetre cubes, can you build more prisms with a volume

of 12 cm3 or with a volume of 14 cm3? Explain your thinking.

e.g., There are more prisms for 12 since it could be 1 × 1 × 12,

or 1 × 2 × 6, or 1 × 3 × 4, but for 14 it can only be 1 × 1 × 14 or 1 × 2 × 7.

9. The formula for the volume of a rectangular prism can be written

as V 5 Abase 3 h.

Why is the formula below another formula for the volume of

a rectangular prism?

V 5 l 3 w 3 h, where l is the length of the base, w is the width of

the base, and h is the height of the prism

FYI

Learning how to

e.g., The area of the base is l × w, so you can just say that instead calculate volumes of

rectangular prisms

of Abase before you multiply by the height. will help you learn

formulas for other

prisms, too.

Volume of Rectangular Prisms, Pathway 3 Leaps and Bounds Copyright © 2012 by Nelson Education Ltd.