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The Height of the triangle is the perpendicular segment from a vertex to the line containing the opposite side. The opposite side is called the Base of the triangle. The terms height and base are also used to represent the segment lengths.

1.
Page 1 of 7

8.4 Area of Triangles

Goal The amount of material needed to

Find the area of triangles. make the sail at the right is

determined by the area of the

Key Words triangular sail.

• height of a triangle The height and base of a triangle are

• base of a triangle used to find its area.

The height of a triangle is the perpendicular segment from a vertex

to the line containing the opposite side. The opposite side is called

the base of the triangle . The terms height and base are also used

to represent the segment lengths.

height height height

base base base

In a right triangle, A height can be A height can be

a leg is a height. inside the triangle. outside the triangle.

As shown in Activity 8.4, the area of a triangle is found using a base

and its corresponding height.

AREA OF A TRIANGLE

1

Words Area 5 }} (base)(height)

2

1

Symbols A 5 }}bh

2 b

Triangles with the Same Area Triangles can have the same area

without necessarily being congruent. For example, all of the

triangles below have the same area but they are not congruent.

8 8 8 8

13 13 13 13

8.4 Area of Triangles 431

8.4 Area of Triangles

Goal The amount of material needed to

Find the area of triangles. make the sail at the right is

determined by the area of the

Key Words triangular sail.

• height of a triangle The height and base of a triangle are

• base of a triangle used to find its area.

The height of a triangle is the perpendicular segment from a vertex

to the line containing the opposite side. The opposite side is called

the base of the triangle . The terms height and base are also used

to represent the segment lengths.

height height height

base base base

In a right triangle, A height can be A height can be

a leg is a height. inside the triangle. outside the triangle.

As shown in Activity 8.4, the area of a triangle is found using a base

and its corresponding height.

AREA OF A TRIANGLE

1

Words Area 5 }} (base)(height)

2

1

Symbols A 5 }}bh

2 b

Triangles with the Same Area Triangles can have the same area

without necessarily being congruent. For example, all of the

triangles below have the same area but they are not congruent.

8 8 8 8

13 13 13 13

8.4 Area of Triangles 431

2.
Page 2 of 7

EXAMPLE 1 Find the Area of a Right Triangle

Find the area of the right triangle.

6 cm

Solution

Use the formula for the area of a triangle. 10 cm

Substitute 10 for b and 6 for h.

1

A 5 }}bh Formula for the area of a triangle

2

1

5 }}(10)(6) Substitute 10 for b and 6 for h.

2

5 30 Simplify.

ANSWER © The triangle has an area of 30 square centimeters.

EXAMPLE 2 Find the Area of a Triangle

Find the area of the triangle.

Solution 8 ft

1

A 5 }}bh Formula for the area of a triangle

2

1

5 }}(8)(5) Substitute 8 for b and 5 for h.

2

5 20 Simplify.

ANSWER © The triangle has an area of 20 square feet.

Visualize It! EXAMPLE 3 Find the Height of a Triangle

To help you determine Find the height of the triangle, given that

the base and height of a

tilted triangle, turn your

its area is 39 square inches. h

book so that the base is 13 in.

horizontal.

Solution

1

h A 5 }}bh Formula for the area of a triangle

2

13 1

in. 39 5 }}(13)h Substitute 39 for A and 13 for b.

2

78 5 13h Multiply each side by 2.

65h Divide each side by 13.

ANSWER © The triangle has a height of 6 inches.

432 Chapter 8 Polygons and Area

EXAMPLE 1 Find the Area of a Right Triangle

Find the area of the right triangle.

6 cm

Solution

Use the formula for the area of a triangle. 10 cm

Substitute 10 for b and 6 for h.

1

A 5 }}bh Formula for the area of a triangle

2

1

5 }}(10)(6) Substitute 10 for b and 6 for h.

2

5 30 Simplify.

ANSWER © The triangle has an area of 30 square centimeters.

EXAMPLE 2 Find the Area of a Triangle

Find the area of the triangle.

Solution 8 ft

1

A 5 }}bh Formula for the area of a triangle

2

1

5 }}(8)(5) Substitute 8 for b and 5 for h.

2

5 20 Simplify.

ANSWER © The triangle has an area of 20 square feet.

Visualize It! EXAMPLE 3 Find the Height of a Triangle

To help you determine Find the height of the triangle, given that

the base and height of a

tilted triangle, turn your

its area is 39 square inches. h

book so that the base is 13 in.

horizontal.

Solution

1

h A 5 }}bh Formula for the area of a triangle

2

13 1

in. 39 5 }}(13)h Substitute 39 for A and 13 for b.

2

78 5 13h Multiply each side by 2.

65h Divide each side by 13.

ANSWER © The triangle has a height of 6 inches.

432 Chapter 8 Polygons and Area

3.
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Area of Triangles

In Exercises 1–3, find the area of the triangle.

1. 2. 3.

8 in. 7 yd 16 cm

15 cm

9 in. 12 yd

4. A triangle has an area of 84 square inches and a height of

14 inches. Find the base.

EXAMPLE 4 Areas of Similar Triangles

a. Find the ratio of the areas of the similar B

triangles. 2

E

b. Find the scale factor of T ABC to TDEF C 4 A

and compare it to the ratio of their areas. 3

F 6 D

Solution T ABC S TDEF

1 1

a. Area of T ABC 5 }}bh 5 }}(4)(2) 5 4 square units

2 2

1 1

Area of TDEF 5 }}bh 5 }}(6)(3) 5 9 square units

2 2

Student Help

Area of T ABC 4

LOOK BACK

Ratio of areas 5 }} 5 }}

Area of TDEF 9

To review scale factor, 2

see p. 366. b. The scale factor of T ABC to TDEF is }}.

3

22 4

The ratio of the areas is the square of the scale factor: }}2 5 }}.

3 9

The relationship in Example 4 is generalized for all similar polygons in

the following theorem.

THEOREM 8.3

Areas of Similar Polygons

Words If two polygons are similar with a scale C

a G

factor of }}, then the ratio of their areas B

b a

a2

is }}2 . F

b b

A D

Symbols If ABCD S EFGH with a scale factor

a Area of ABCD a2

of }}, then }} 5 }}2 . E H

b Area of EFGH b

8.4 Area of Triangles 433

Area of Triangles

In Exercises 1–3, find the area of the triangle.

1. 2. 3.

8 in. 7 yd 16 cm

15 cm

9 in. 12 yd

4. A triangle has an area of 84 square inches and a height of

14 inches. Find the base.

EXAMPLE 4 Areas of Similar Triangles

a. Find the ratio of the areas of the similar B

triangles. 2

E

b. Find the scale factor of T ABC to TDEF C 4 A

and compare it to the ratio of their areas. 3

F 6 D

Solution T ABC S TDEF

1 1

a. Area of T ABC 5 }}bh 5 }}(4)(2) 5 4 square units

2 2

1 1

Area of TDEF 5 }}bh 5 }}(6)(3) 5 9 square units

2 2

Student Help

Area of T ABC 4

LOOK BACK

Ratio of areas 5 }} 5 }}

Area of TDEF 9

To review scale factor, 2

see p. 366. b. The scale factor of T ABC to TDEF is }}.

3

22 4

The ratio of the areas is the square of the scale factor: }}2 5 }}.

3 9

The relationship in Example 4 is generalized for all similar polygons in

the following theorem.

THEOREM 8.3

Areas of Similar Polygons

Words If two polygons are similar with a scale C

a G

factor of }}, then the ratio of their areas B

b a

a2

is }}2 . F

b b

A D

Symbols If ABCD S EFGH with a scale factor

a Area of ABCD a2

of }}, then }} 5 }}2 . E H

b Area of EFGH b

8.4 Area of Triangles 433

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Page 4 of 7

8.4 Exercises

Guided Practice

Vocabulary Check 1. What are the measures of the base and 13 ft

the height of the shaded triangle at the 5 ft

right?

3 ft 9 ft

Skill Check The triangle has a horizontal base of 15 units and a height of 7 units.

Sketch the triangle and label its base and its height.

2. 3. 4.

Practice and Applications

Extra Practice Area of a Right Triangle Find the area of the right triangle.

See p. 690. 5. 6. 7.

7 cm 6 ft 3 yd 5 yd

12 cm 7 ft

Finding Area In Exercises 8–13, find the area of the triangle.

8. 9. 7 yd 10.

4m

8 mm

4 yd

9m

14 mm

11. 12. 13.

12 in. 9 ft 14 ft

4 cm

12 in.

5 cm

14. You be the Judge In the triangle

Homework Help at the right, Trisha says the base is 15

and the height is 4. Luis says that the

Example 1: Exs. 5–7 base is 5 and the height is 12. Who is 15 12

Example 2: Exs. 8–13 right? Explain your reasoning.

Example 3: Exs. 16–20 4

Example 4: Exs. 26, 27

5

434 Chapter 8 Polygons and Area

8.4 Exercises

Guided Practice

Vocabulary Check 1. What are the measures of the base and 13 ft

the height of the shaded triangle at the 5 ft

right?

3 ft 9 ft

Skill Check The triangle has a horizontal base of 15 units and a height of 7 units.

Sketch the triangle and label its base and its height.

2. 3. 4.

Practice and Applications

Extra Practice Area of a Right Triangle Find the area of the right triangle.

See p. 690. 5. 6. 7.

7 cm 6 ft 3 yd 5 yd

12 cm 7 ft

Finding Area In Exercises 8–13, find the area of the triangle.

8. 9. 7 yd 10.

4m

8 mm

4 yd

9m

14 mm

11. 12. 13.

12 in. 9 ft 14 ft

4 cm

12 in.

5 cm

14. You be the Judge In the triangle

Homework Help at the right, Trisha says the base is 15

and the height is 4. Luis says that the

Example 1: Exs. 5–7 base is 5 and the height is 12. Who is 15 12

Example 2: Exs. 8–13 right? Explain your reasoning.

Example 3: Exs. 16–20 4

Example 4: Exs. 26, 27

5

434 Chapter 8 Polygons and Area

5.
Page 5 of 7

15. Visualize It! Draw three different triangles that each have an

area of 24 square units.

Using Algebra In Exercises 16–18, A gives the area of the

triangle. Find the missing measure.

16. A 5 22 ft2 17. A 5 63 cm2 18. A 5 80 m2

b

4 ft h b

14 cm

16 m

19. Finding the Height A triangle has an area of 78 square inches and

a base of 13 inches. Find the height.

20. Finding the Base A triangle has an area of 135 square meters and

a height of 9 meters. Find the base.

Tiles In Exercises 21 and 22, use the diagram of the tile pattern.

2 in.

4 in.

21. Find the area of one triangular tile.

22. The tiles are being used to make a rectangular border that is

4 inches high and 48 inches long. How many tiles are needed

for the border? (Hint: Start by finding the area of the border.)

Complex Polygons Find the area of the polygon by using the

triangles and rectangles shown.

23. 24. 25. 8 cm

5 ft

4m

12 ft 6 cm

7 ft

4m

5 ft 5 cm

7m

Areas of Similar Triangles In Exercises 26 and 27, the triangles are

similar. Find the scale factor of TPQR to TXYZ. Then find the ratio of

their areas.

26. P 27. P

3 Y

Y 6

P 5 R

6 P 9 R 8

X 10 Z X 12 Z

8.4 Area of Triangles 435

15. Visualize It! Draw three different triangles that each have an

area of 24 square units.

Using Algebra In Exercises 16–18, A gives the area of the

triangle. Find the missing measure.

16. A 5 22 ft2 17. A 5 63 cm2 18. A 5 80 m2

b

4 ft h b

14 cm

16 m

19. Finding the Height A triangle has an area of 78 square inches and

a base of 13 inches. Find the height.

20. Finding the Base A triangle has an area of 135 square meters and

a height of 9 meters. Find the base.

Tiles In Exercises 21 and 22, use the diagram of the tile pattern.

2 in.

4 in.

21. Find the area of one triangular tile.

22. The tiles are being used to make a rectangular border that is

4 inches high and 48 inches long. How many tiles are needed

for the border? (Hint: Start by finding the area of the border.)

Complex Polygons Find the area of the polygon by using the

triangles and rectangles shown.

23. 24. 25. 8 cm

5 ft

4m

12 ft 6 cm

7 ft

4m

5 ft 5 cm

7m

Areas of Similar Triangles In Exercises 26 and 27, the triangles are

similar. Find the scale factor of TPQR to TXYZ. Then find the ratio of

their areas.

26. P 27. P

3 Y

Y 6

P 5 R

6 P 9 R 8

X 10 Z X 12 Z

8.4 Area of Triangles 435

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Page 6 of 7

Area of a Regular Octagon In Exercises 28–30, use the regular

octagon at the right.

Geology B C

28. Find the area of TGXF in the octagon.

A D

29. Copy the diagram. To form congruent

X

triangles, connect the following pairs of

vertices: A and E, B and F, C and G, D H E

and H. How many triangles are formed?

G 4 F

30. What is the area of the octagon? Explain.

31. Rock Formations Many basaltic columns are hexagonal. The

top of one of these columns is a regular hexagon as shown

below. Find its area. (Another photograph of basaltic columns

is on page 408.)

ROCK FORMATIONS

Geologists learn about the

structure of the earth by

studying rock formations such 7.8 in.

as the basaltic columns at the

Giant’s Causeway in Ireland

pictured above.

9 in.

Application Links

CLASSZONE.COM

EXAMPLE Using the Pythagorean Theorem

Find the area of the triangle. 13

5

b

Solution

First, find the base. Use the Pythagorean

Theorem to find the value of b.

Student Help (hypotenuse)2 5 (leg)2 1 (leg)2 Pythagorean Theorem

2 2 2

LOOK BACK (13) 5 (5) 1 (b) Substitute.

To review the 2

Pythagorean Theorem, 169 5 25 1 b Simplify.

see pp. 192 and 193.

144 5 b2 Subtract 25 from each side.

12 5 b Find the positive square root.

Use 12 as the base in the formula for the area of a triangle.

1 1

A 5 }}bh 5 }}(12)(5) 5 30 square units

2 2

Using the Pythagorean Theorem Find the area of the triangle.

32. 33. 34.

26

a 20 12

6 10

24

b

b

436 Chapter 8 Polygons and Area

Area of a Regular Octagon In Exercises 28–30, use the regular

octagon at the right.

Geology B C

28. Find the area of TGXF in the octagon.

A D

29. Copy the diagram. To form congruent

X

triangles, connect the following pairs of

vertices: A and E, B and F, C and G, D H E

and H. How many triangles are formed?

G 4 F

30. What is the area of the octagon? Explain.

31. Rock Formations Many basaltic columns are hexagonal. The

top of one of these columns is a regular hexagon as shown

below. Find its area. (Another photograph of basaltic columns

is on page 408.)

ROCK FORMATIONS

Geologists learn about the

structure of the earth by

studying rock formations such 7.8 in.

as the basaltic columns at the

Giant’s Causeway in Ireland

pictured above.

9 in.

Application Links

CLASSZONE.COM

EXAMPLE Using the Pythagorean Theorem

Find the area of the triangle. 13

5

b

Solution

First, find the base. Use the Pythagorean

Theorem to find the value of b.

Student Help (hypotenuse)2 5 (leg)2 1 (leg)2 Pythagorean Theorem

2 2 2

LOOK BACK (13) 5 (5) 1 (b) Substitute.

To review the 2

Pythagorean Theorem, 169 5 25 1 b Simplify.

see pp. 192 and 193.

144 5 b2 Subtract 25 from each side.

12 5 b Find the positive square root.

Use 12 as the base in the formula for the area of a triangle.

1 1

A 5 }}bh 5 }}(12)(5) 5 30 square units

2 2

Using the Pythagorean Theorem Find the area of the triangle.

32. 33. 34.

26

a 20 12

6 10

24

b

b

436 Chapter 8 Polygons and Area

7.
Page 7 of 7

Standardized Test 35. Multiple Choice Given that the area of the triangle is 99 square

Practice meters, what is the height of the triangle?

A 5 99 m2

X

A 4.5 m B 9m

X h

X

C 11 m D 22 m

X

22 m

Mixed Review Trapezoids Find the value of x in the trapezoid. (Lesson 6.5)

36. 6 37. 20 38. 18

x 16 21

x x

12

Algebra Skills Naming Coordinates Give the coordinates of the point.

(Skills Review, p. 664)

A y

39. A 40. B

C

41. C 42. D B

1

43. E 44. F 1 x

D E

F

Quiz 2

Find the area of the polygon. (Lessons 8.3, 8.4)

1. 2. 3. 5m

3 in.

7 in. 8m

12 cm 3m

7m

4. 5. 14 mm 6.

12 yd

14 ft

8 mm

16 yd

22 ft

In Exercises 7–9, A gives the area of the polygon. Find the missing

measure. (Lessons 8.3, 8.4)

7. A 5 48 in.2 8. A 5 90 m2 9. A 5 63 cm2

9 cm

6 in. h

b 15 m b

8.4 Area of Triangles 437

Standardized Test 35. Multiple Choice Given that the area of the triangle is 99 square

Practice meters, what is the height of the triangle?

A 5 99 m2

X

A 4.5 m B 9m

X h

X

C 11 m D 22 m

X

22 m

Mixed Review Trapezoids Find the value of x in the trapezoid. (Lesson 6.5)

36. 6 37. 20 38. 18

x 16 21

x x

12

Algebra Skills Naming Coordinates Give the coordinates of the point.

(Skills Review, p. 664)

A y

39. A 40. B

C

41. C 42. D B

1

43. E 44. F 1 x

D E

F

Quiz 2

Find the area of the polygon. (Lessons 8.3, 8.4)

1. 2. 3. 5m

3 in.

7 in. 8m

12 cm 3m

7m

4. 5. 14 mm 6.

12 yd

14 ft

8 mm

16 yd

22 ft

In Exercises 7–9, A gives the area of the polygon. Find the missing

measure. (Lessons 8.3, 8.4)

7. A 5 48 in.2 8. A 5 90 m2 9. A 5 63 cm2

9 cm

6 in. h

b 15 m b

8.4 Area of Triangles 437