10-4 area of triangles -...

10-4 Area of Triangles

BIOSPHERE The structure of the different sections in the Biosphere 2 complex in Tucson, Arizona, are made of interlocking triangles that are all the same size.

1. Compare the two outlined triangles.

2. What figure is formed by the two triangles?

3. MAKE A CONJECTURE Describe the relationship that exists between the area of one triangle and the area of the parallelogram.

A parallelogram can be formed by two congruent triangles. Since congruent triangles have the same area, the area of a triangle is one half the area of the parallelogram.

height (h)

base (b)

The base of a triangle can be any one of its sides. The height is the shortest distance from a base tothe opposite vertex.

Find the Area of a Triangle

1

Find the area of each triangle.

base

height

By counting, you find that the measure of the base is 6 units and the height is 4 units.

540 Chapter 10 Measurement: Perimeter, Area, and Volume

MAIN IDEAFind the areas of triangles.

Math Online

glencoe.com• Concepts in Motion• Extra Examples• Personal Tutor• Self-Check Quiz

Key ConceptArea of a Triangle

Words The area A of a triangle is one half the product of the base b and its height h.

Model h

bSymbols A = 1 _

2 bh or A = bh _ 2

Epigraphics, Inc. Middle School Math ©2009, Course 1

1st pass 2nd pass 3rd pass 4th pass 5th pass Printer PDF

539_544_C10_L04_874042.indd 540539_544_C10_L04_874042.indd 540 9/21/07 12:15:07 PM9/21/07 12:15:07 PM

A = 1 _ 2 bh Area of a triangle

A = 1 _ 2 (6)(4) Replace b with 6 and h with 4.

A = 1 _ 2 (24) Multiply.

A = 12 Multiply.

The area of the triangle is 12 square units.

2 12.1 m

6.4 m


A = 1 _ 2 (12.1)(6.4) Replace b with 12.1 and h with 6.4.

A = 1 _ 2 (77.44) Multiply.

A = 38.72 Divide. 1 _ 2 (77.44) = 77.44 ÷ 2, or 38.72

The area of the triangle is 38.72 square meters.

a. b. 9 ft

7 ft

3 TENTS The front of a two-person camping tent has the dimensions shown. How much material was used to make the front of the tent?


A = 1 _ 2 (5)(3) Replace b with 5 and h

with 3.

A = 1 _ 2 (15) or 7.5 Multiply.

The front of the tent has an area of 7.5 square feet.

c. SNACKS A triangular cracker has a height of 4 centimeters and a

base of 5 centimeters. Find the area of the cracker.

5 ft

3 ft

Check for Reasonableness Check for Reasonableness To estimate the area of the triangle in Example 2, round the base to 12 meters and the height to 6 meters.

The area is then 12 × 6 _

2 or

36 square meters. Since 38.72 is close to 36, the answer is reasonable.

Mental Math Mental Math You can use mental math to multiply

1 _ 2 (6)(4). Think: Half of 6 is

3, and 3 × 4 is 12.

Lesson 10-4 Area of Triangles 541




Examples 1, 2(pp. 540–541)


1. 2.

8 ft

12 ft

3. 11.25 m

15.6 m

Example 3(p. 541)

4. CRAFTS Consuela made a triangular paper

10 cm

9 cmbox as shown. What is the area of the top

of the box?


For Exercises

5, 67–1213, 14

See Examples

123

HOMEWORK HELPHELP 5. 6. 7.

9 in.

10 in.

8.

16 cm

24.8 cm

9. 7 m

25 m

10.

36 ft41 ft1

2

11. height: 14 in., base: 35 in. 12. height: 27 cm, base: 19 cm

13. ROOFING Ansley is going to help his father shingle

7 yd

4 ydthe roof of their house. What is the area of the triangular portion of one end of the roof to be shingled?

14. ARCHITECTURE An architect plans on designing a building on a triangular plot of land. If the base of the triangle is 100.8 feet and the height is 96.3 feet, find the available floor area the architect has to design the building.

15. FLOWER BEDS A flower bed in a parking lot is 3 yd

2 yd

shaped like a triangle as shown. Find the area of the flower bed in square feet. If one bag of topsoil covers 10 square feet, how many bags are needed to cover this flower bed?





Area of TrianglesBase

(units)Height (units)

Area (units2)

5 2 55 4 105 6 155 8 205 n �

16. ALGEBRA The table at the right shows the areas

of a triangle where the base of the triangle stays

the same but the height changes. Write an

algebraic expression that can be used to find the

area of a triangle that has a base of 5 units and a

height of n units.

17. REASONING Which is smaller, a triangle with an area of 1 square foot or a triangle with an area of 64 square inches?

18. FLAGS What is the area of the

30 in.

5 ft

3 ft

triangle on the flag of the Philippines at the right?

COMPOSITE FIGURES Find the perimeter and area of each figure.

19.

5 in.

4 in.

2 in.

5 in.

20. 9 mm6 mm

9 mm

10 mm 10 mm8 mm8 mm

6 mm

H.O.T. Problems 21. FIND THE ERROR Dolores and Demetrius are

20 m

b m

finding the base of the triangle shown. Its area is 148.5 square meters. Who is correct? Explain.

100 = 1 _

2 (b)(20)

100 = 10b 10 = b

100 = (b)20

100 = 20b 5 = b

Dolores Demetrius

CHALLENGE For Exercises 22–25, use the information below.

All the triangles and squares in the quilt pattern

12 in.

12 in.

shown are congruent.

22. Find the measure of the base and height of one of the triangles.

23. Calculate the area of one triangle and then find the area of all the triangles.

24. Calculate the area of one of the smaller squares and then find the area of all of the smaller squares.

25. What is the total area of the figure? Is your answer reasonable?

Lesson 10-4 Area of Triangles 543

See pages 699, 715.

EXTRA PRACTICEPRACTICE




30. Find the area of a parallelogram with base 15 inches and height 10 inches. (Lesson 10-3)

31. Find the circumference of a circle with a radius of 5 meters. Round to the nearest tenth. (Lesson 10-2)

32. IDENTIFICATION Measure the length and width of a student ID card or library card to the nearest eighth inch. Then find the perimeter of the card. (Lessons 1-9 and 10-1)

33. PREREQUISITE SKILL A bookstore arranges its best-seller books in the front window. In how many ways can four best-seller books be arranged in a row? Use the act it out strategy. (Lesson 5-3)

26. REASONING If two triangles have an area of 24 square feet, do they always have the same base and height? Use a model to explain your answer.

27. MATHWRITING IN Draw a triangle and label its base and height. Draw another triangle that has the same base, but a height twice that of the first triangle. Find the area of each triangle. Then write a ratio that expresses the area of the first triangle to the area of the second triangle.


28. The table below shows the areas of a triangle where the height of the triangle stays the same but the base changes.

Areas of TrianglesHeight (units)

Base (units)

Area (square units)

7 2 7

7 3 10 1 _ 2

7 4 14

7 5 17 1 _ 2

7 x ?

Which expression can be used to find the area of a triangle that has a height of 7 units and a base of x units?

A 7x C 7 _ 2

B 7x _ 2 D x _

2

29. Norma cut a triangle out of construction paper for an art project.

13 cm

h cm

The area of the triangle is 84.5 square centimeters. What is the height of the triangle?

F 6.5 cm

G 13 cm

H 26 cm

J 169 cm




10-4 area of triangles -...

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