8.4 Area of Triangles

Page 1 of 7

8.4

Goal

Find the area of triangles.

Key Words

? height of a triangle

? base of a triangle

Area of Triangles

The amount of material needed to

make the sail at the right is

determined by the area of the

triangular sail.

The height and base of a triangle are

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

base

height

base

In a right triangle,

a leg is a height.

base

A height can be

inside the triangle.

A height can be

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

Words

1

2

Area 5 }} (base)(height)

Symbols

1

2

A 5 }}bh

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

13

13

13

8

13

8.4

Area of Triangles

431

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.

Substitute 10 for b and 6 for h.

1

2

A 5 }}bh

10 cm

Formula for the area of a triangle

1

2

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

5 30

ANSWER

Simplify.

? 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

2

Formula for the area of a triangle

1

2

5 }}(8)(5)

Substitute 8 for b and 5 for h.

5 20

Simplify.

ANSWER

? The triangle has an area of 20 square feet.

Visualize It!

EXAMPLE

To help you determine

the base and height of a

tilted triangle, turn your

book so that the base is

horizontal.

Find the height of the triangle, given that

its area is 39 square inches.

3

Find the Height of a Triangle

h

13 in.

Solution

1

2

Formula for the area of a triangle

39 5 }}(13)h

1

2

Substitute 39 for A and 13 for b.

78 5 13h

Multiply each side by 2.

A 5 }}bh

h

13

in.

65h

ANSWER

432

Chapter 8

Polygons and Area

Divide each side by 13.

? The triangle has a height of 6 inches.

Page 3 of 7

Area of Triangles

In Exercises 1¨C3, find the area of the triangle.

1.

2.

3.

8 in.

16 cm

7 yd

15 cm

12 yd

9 in.

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

2

triangles.

b. Find the scale factor of T ABC to TDEF

C

E

A

4

3

and compare it to the ratio of their areas.

F

6

D

T ABC S TDEF

Solution

1

2

1

2

1

2

1

2

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

Student Help

LOOK BACK

To review scale factor,

see p. 366.

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

Area of T ABC

Area of TDEF

4

9

Ratio of areas 5 }} 5 }}

2

3

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

22

3

4

9

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

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

the following theorem.

THEOREM 8.3

Areas of Similar Polygons

Words

C

If two polygons are similar with a scale

a

factor of }}, then the ratio of their areas

b

a2

is }}2 .

b

G

B

a

F

A

Symbols

b

D

If ABCD S EFGH with a scale factor

a

b

Area of ABCD

Area of EFGH

a2

b

of }}, then }} 5 }}2 .

E

8.4

Area of Triangles

H

433

Page 4 of 7

8.4 Exercises

Guided Practice

Vocabulary Check

1. What are the measures of the base and

the height of the shaded triangle at the

right?

Skill Check

13 ft

5 ft

3 ft

9 ft

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 cm

7.

6 ft

12 cm

3 yd

5 yd

7 ft

Finding Area In Exercises 8¨C13, find the area of the triangle.

8.

9.

10.

7 yd

4m

8 mm

4 yd

9m

14 mm

11.

12.

13.

12 in.

9 ft

14 ft

4 cm

12 in.

5 cm

14.

Homework Help

Example 1:

Example 2:

Example 3:

Example 4:

434

Exs. 5¨C7

Exs. 8¨C13

Exs. 16¨C20

Exs. 26, 27

Chapter 8

You be the Judge

In the triangle

at the right, Trisha says the base is 15

and the height is 4. Luis says that the

base is 5 and the height is 12. Who is

right? Explain your reasoning.

Polygons and Area

15

4

5

12

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¨C18, 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

b

h

4 ft

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.

4m

8 cm

25.

5 ft

12 ft

6 cm

7 ft

4m

5 cm

5 ft

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.

27.

P

3

Y

P

Y

5

6

R

P

6

X

P

10

9

R

X

Z

8.4

8

12

Area of Triangles

Z

435

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