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Finding Area, Perimeter, and Circumference of circle, square, and other polygons. Area refers to the space occupied by a shape or an object or a surface. Perimeter refers to the measure of the length of the outline or boundary of a shape, an object, or a surface.

1.
FINDING AREA, PERIMETER, AND CIRCUMFERENCE

Area, perimeter, and circumference are all measures of two-dimensional shapes. These

are things you can think of as flat: a football field, a piece of paper, or a pizza. You're

probably not interested in how high they are, but you might want to know their:

● Perimeter or Circumference. This is the total length of a shape's outline. If you

built a fence around its edge, how long would that fence be? If you walked

around the edges of this area, how far would you have gone? The length of a

straight-sided shape's outline is called its perimeter, and the length of a circle's

outline is called its circumference.

● Area. This is the total amount of space inside a shape's outline. If you wanted to

paint a wall or irrigate a circular field, how much space would you have to cover?

1. The perimeter of any triangle is the sum of its sides: a + b + c

perimeter = a + b + c

c 5 P=a+b+c

a 3 P=3+5+4

P = 12

b 4

2. The area of any triangle is half its base times its height.

area = 1/2 bh

A = 1/2 bh

h 3 A = 1/2 * 3 * 4

A = 1/2 * 12

b 4 A=6

It doesn't matter which of the triangle's short legs is the "base" and which is the

"height": you get the same solution either way.

A = 1/2 bh

h 4 A = 1/2 * 4 * 3

A=2*3

A=6

b 3

Area, perimeter, and circumference are all measures of two-dimensional shapes. These

are things you can think of as flat: a football field, a piece of paper, or a pizza. You're

probably not interested in how high they are, but you might want to know their:

● Perimeter or Circumference. This is the total length of a shape's outline. If you

built a fence around its edge, how long would that fence be? If you walked

around the edges of this area, how far would you have gone? The length of a

straight-sided shape's outline is called its perimeter, and the length of a circle's

outline is called its circumference.

● Area. This is the total amount of space inside a shape's outline. If you wanted to

paint a wall or irrigate a circular field, how much space would you have to cover?

1. The perimeter of any triangle is the sum of its sides: a + b + c

perimeter = a + b + c

c 5 P=a+b+c

a 3 P=3+5+4

P = 12

b 4

2. The area of any triangle is half its base times its height.

area = 1/2 bh

A = 1/2 bh

h 3 A = 1/2 * 3 * 4

A = 1/2 * 12

b 4 A=6

It doesn't matter which of the triangle's short legs is the "base" and which is the

"height": you get the same solution either way.

A = 1/2 bh

h 4 A = 1/2 * 4 * 3

A=2*3

A=6

b 3

2.
1. A square is a kind of rectangle, and the perimeter of any rectangle is the sum of its

four sides. Since all sides of a square are the same,

perimeter = 4s

P = 4s

s 3 P=4*3

P = 12

s 3

2. The area of a square is equal to any one of its sides times any other: s * s . Since

that's the same as s squared,

area = s2

A = s2

s 3 A = 32

A=3*3

s 3 A=9

1. The perimeter of a rectangle is the sum of its four sides. Since a rectangle has two

equal short sides (width, w ) and two equal long sides (length, l ),

perimeter = 2l + 2w

P = 2l + 2w

w 3 P = (2 * 7) + (2 * 3)

P = 14 + 6

l 7 P = 20

2. The area of a rectangle is equal to its length times its width.

area = l * w

A=l*w

w 3 A=3*7

A = 21

l 7

four sides. Since all sides of a square are the same,

perimeter = 4s

P = 4s

s 3 P=4*3

P = 12

s 3

2. The area of a square is equal to any one of its sides times any other: s * s . Since

that's the same as s squared,

area = s2

A = s2

s 3 A = 32

A=3*3

s 3 A=9

1. The perimeter of a rectangle is the sum of its four sides. Since a rectangle has two

equal short sides (width, w ) and two equal long sides (length, l ),

perimeter = 2l + 2w

P = 2l + 2w

w 3 P = (2 * 7) + (2 * 3)

P = 14 + 6

l 7 P = 20

2. The area of a rectangle is equal to its length times its width.

area = l * w

A=l*w

w 3 A=3*7

A = 21

l 7

3.
Like squares and rectangles, parallelograms are quadrilaterals: they have four sides

and four interior angles. In a parallelogram those angles are not right angles, but

the opposite sides must still be parallel to each other.

1. The perimeter of a parallelogram is the sum of its four sides. Since a parallelogram

has two equal short sides (width, w ) and two equal long sides (length, l ),

perimeter = 2l + 2w

P = 2l + 2w

w 4 P = (2*5) + (2 * 4)

P = 10 +8

l 5 P = 18

2. The area of a parallelogram is equal to its base (another name for length) times

its height. Its height is not the same as its width: height is measured by a vertical line

perpendicular (at right angles to) the base.

area = b * h

A=b*h

h 3 A=3*5

A = 15

b 5

A trapezoid is also a quadrilateral: it has four sides, but only two are parallel.

1. The perimeter of a trapezoid is the sum of its four sides.

perimeter = a + b + c + d

b 2

P=a+b+c+d

a c 4 3 P=2+3+4+5

P = 14

d 5

2. To find the area of a trapezoid, we use its two bases and its height:

area = 1/2 (b1 + b2) (h) A = 1/2 (b 1 + b 2 ) * (h)

b1 2 A = 1/2 (2 + 5) * 3

A = 1/2 * 7 * 3

h 3 A = 1/2 * 21

A = 10.5

b2 5

and four interior angles. In a parallelogram those angles are not right angles, but

the opposite sides must still be parallel to each other.

1. The perimeter of a parallelogram is the sum of its four sides. Since a parallelogram

has two equal short sides (width, w ) and two equal long sides (length, l ),

perimeter = 2l + 2w

P = 2l + 2w

w 4 P = (2*5) + (2 * 4)

P = 10 +8

l 5 P = 18

2. The area of a parallelogram is equal to its base (another name for length) times

its height. Its height is not the same as its width: height is measured by a vertical line

perpendicular (at right angles to) the base.

area = b * h

A=b*h

h 3 A=3*5

A = 15

b 5

A trapezoid is also a quadrilateral: it has four sides, but only two are parallel.

1. The perimeter of a trapezoid is the sum of its four sides.

perimeter = a + b + c + d

b 2

P=a+b+c+d

a c 4 3 P=2+3+4+5

P = 14

d 5

2. To find the area of a trapezoid, we use its two bases and its height:

area = 1/2 (b1 + b2) (h) A = 1/2 (b 1 + b 2 ) * (h)

b1 2 A = 1/2 (2 + 5) * 3

A = 1/2 * 7 * 3

h 3 A = 1/2 * 21

A = 10.5

b2 5

4.
To find a circle's circumference or area, you first need to know either its

radius: r , the distance from its center to any point on its outer edge, or its

diameter: d , the length of a straight line through the circle's center that touches

any two points on the outer edge.

A circle's radius is always exactly half its diameter.

r d

1. The circumference of any circle equals two times its radius multiplied by pi

(π, approximately 3.14). We can also say it equals pi times its diameter.

circumference = 2 π r OR πd

C=2 π*3

r 3 C=6* π

C ≈ 18.84

Because 3.14 is only an approximate value for pi, we replace the "equals" sign (=)

with the "approximately equals" sign ( ≈). For accuracy, some teachers prefer

to use the symbol: the circumference of this circle is 6 π .

2. To find the area of a circle, square its radius and multiply the result by pi .

2

area = πr

A = πr 2

r 3 A = 32 * π

A = (3 * 3) * π

A=9* π

A = 9π or ≈ 28.26

Created Fall 2012

Pocatello ISU Math Center Idaho Falls

REND 327 Student Success Center CHE 220

208-282-3662 www.isu.edu/success/math 208-282-7925

radius: r , the distance from its center to any point on its outer edge, or its

diameter: d , the length of a straight line through the circle's center that touches

any two points on the outer edge.

A circle's radius is always exactly half its diameter.

r d

1. The circumference of any circle equals two times its radius multiplied by pi

(π, approximately 3.14). We can also say it equals pi times its diameter.

circumference = 2 π r OR πd

C=2 π*3

r 3 C=6* π

C ≈ 18.84

Because 3.14 is only an approximate value for pi, we replace the "equals" sign (=)

with the "approximately equals" sign ( ≈). For accuracy, some teachers prefer

to use the symbol: the circumference of this circle is 6 π .

2. To find the area of a circle, square its radius and multiply the result by pi .

2

area = πr

A = πr 2

r 3 A = 32 * π

A = (3 * 3) * π

A=9* π

A = 9π or ≈ 28.26

Created Fall 2012

Pocatello ISU Math Center Idaho Falls

REND 327 Student Success Center CHE 220

208-282-3662 www.isu.edu/success/math 208-282-7925