Objective - Mrs. Meyer's Math SiteProblem 1 Lesson 10-3 Areas of Regular Polygons 629 Objective To find the area of a regular polygon 10-3 Areas of Regular Polygons You want to build

Problem 1

Lesson 10-3 Areas of Regular Polygons 629

Objective To find the area of a regular polygon

10-3 Areas of Regular Polygons

You want to build a koi pond. For the border, you plan to use 3-ft-long pieces of wood. You have 12 pieces that you can connect together at any angle, including a straight angle. If you want to maximize the area of the pond, in what shape should you arrange the pieces? Explain your reasoning.

The Solve It involves the area of a polygon.

Essential Understanding The area of a regular polygon is related to the distance from the center to a side.

You can circumscribe a circle about any regular polygon. The center of a regular polygon is the center of the circumscribed circle. The radius of a regular polygon is the distance from the center to a vertex. The apothem is the perpendicular distance from the center to a side.

Center

Radius

Apothem

hsm11gmse_1003_t09346.ai

Lesson Vocabulary

•radius of a regular polygon

•apothem

LessonVocabulary

Finding Angle Measures

The figure at the right is a regular pentagon with radii and an apothem drawn. What is the measure of each numbered angle?

m∠1 = 3605 = 72 Divide 360 by the number of sides.

m∠2 = 12m∠1 The apothem bisects the vertex angle of the

isosceles triangle formed by the radii.

= 12(72) = 36

90 + 36 + m∠3 = 180 The sum of the measures of the angles of a triangle is 180.

m∠3 = 54

m∠1 = 72, m∠2 = 36, and m∠3 = 54.

1. At the right, a portion of a regular octagon has radii and an apothem drawn. What is the measure of each numbered angle?


3

2 1


2

3

1Got It?

How do you know the radii make isosceles triangles?Since the pentagon is a regular polygon, the radii are congruent. So, the triangle made by two adjacent radii and a side of the polygon is an isosceles triangle.

Solve a simpler problem. Try using fewer sides to see what happens.

G-MG.A.1 Use geometric shapes, their measures, and their properties to describe objects. Also G-CO.D.13

MP 1, MP 3, MP 4, MP 6, MP 7

MATHEMATICAL PRACTICES

Common Core State Standards

Problem 2

630 Chapter 10 Area

Suppose you have a regular n-gon with side s. The radii divide the figure into n congruent isosceles triangles. By Postulate 10-1, the areas of the isosceles triangles are equal. Each triangle has a height of a and a base of length s, so the area of each triangle is 12as.

Since there are n congruent triangles, the area of the n-gon is A = n # 1

2as. The perimeter p of the n-gon is the number of sides n times the length of a side s, or ns. By substitution, the area can be expressed as A = 1

2ap.

Finding the Area of a Regular Polygon

What is the area of the regular decagon at the right?

Step 1 Find the perimeter of the regular decagon.

p = ns Use the formula for the perimeter of an n-gon.

= 10(8) Substitute 10 for n and 8 for s.

= 80 in.

Step 2 Find the area of the regular decagon.

A = 12ap Use the formula for the area of a regular polygon.

= 12(12.3)(80) Substitute 12.3 for a and 80 for p.

= 492

The regular decagon has an area of 492 in.2.

2. a. What is the area of a regular pentagon with an 8-cm apothem and 11.6-cm sides?

b. Reasoning If the side of a regular polygon is reduced to half its length, how does the perimeter of the polygon change? Explain.

Postulate 10-1

If two figures are congruent, then their areas are equal.


s

a

Theorem 10-6 Area of a Regular Polygon

The area of a regular polygon is half the product of the apothem and the perimeter.

A = 12ap


a

p


12.3 in.

8 in.

Got It?

What do you know about the regular decagon?A decagon has 10 sides, so n = 10. From the diagram, you know that the apothem a is 12.3 in., and the side length s is 8 in.

Problem 3


Using Special Triangles to Find Area

Zoology A honeycomb is made up of regular hexagonal cells. The length of a side of a cell is 3 mm. What is the area of a cell?

Step 1 Find the apothem.

The radii form six 60° angles at the center, so you can use a 30°-60°-90° triangle to find the apothem.

a = 1.513 longer leg = 13 # shorter leg

Step 2 Find the perimeter.

p = ns Use the formula for the perimeter of an n-gon.

= 6(3) Substitute 6 for n and 3 for s.

= 18 mm

Step 3 Find the area.

A = 12ap Use the formula for the area of a regular polygon.

= 12(1.513) (18) Substitute 1.513 for a and 18 for p.

≈ 23.3826859 Use a calculator.

The area is about 23 mm2.

3. The side of a regular hexagon is 16 ft. What is the area of the hexagon? Round your answer to the nearest square foot.

Draw a diagram to help find the apothem. Then use the area formula for a regular polygon.

The apothemYou know the length of a side, which you can use to find the perimeter.


a

1.5 mm

3 mm30� 60�

Got It?

Do you know HOW?What is the area of each regular polygon? Round your answer to the nearest tenth.

1. 2.

3. 4.

Do you UNDERSTAND? 5. Vocabulary What is the difference between a radius

and an apothem?

6. What is the relationship between the side length and the apothem in each figure?

a. a square b. a regular hexagon c. an equilateral triangle

7. Error Analysis Your friend says you can use special triangles to find the apothem of any regular polygon. What is your friend’s error? Explain.


5 in.


3 ft


2 m


4V3

Lesson CheckMATHEMATICAL PRACTICES

STEM

632 Chapter 10 Area

Practice and Problem-Solving Exercises

Each regular polygon has radii and apothem as shown. Find the measure of each numbered angle.

8. 9. 10.

Find the area of each regular polygon with the given apothem a and side length s.

11. pentagon, a = 24.3 cm, s = 35.3 cm 12. 7-gon, a = 29.1 ft, s = 28 ft

13. octagon, a = 60.4 in., s = 50 in. 14. nonagon, a = 27.5 in., s = 20 in.

15. decagon, a = 19 m, s = 12.3 m 16. dodecagon, a = 26.1 cm, s = 14 cm

Find the area of each regular polygon. Round your answer to the nearest tenth.

17. 18. 19.

20. Art You are painting a mural of colored equilateral triangles. The radius of each triangle is 12.7 in. What is the area of each triangle to the nearest square inch?

Find the area of each regular polygon with the given radius or apothem. If your answer is not an integer, leave it in simplest radical form.

21. 22.

23. 24. 25.

Find the measures of the angles formed by (a) two consecutive radii and (b) a radius and a side of the given regular polygon.

26. pentagon 27. octagon 28. nonagon 29. dodecagon

PracticeA See Problem 1.


3

21


4

5

6


7

8

9

See Problem 2.

See Problem 3.


18 ft


8 in.


6 m


s

s2

12.7 in.

30�


6 cm


8V3 in.


6V3 m 5 m


4 in.


ApplyB

MATHEMATICAL PRACTICES


30. Satellites One of the smallest space satellites ever developed has the shape of a pyramid. Each of the four faces of the pyramid is an equilateral triangle with sides about 13 cm long. What is the area of one equilateral triangular face of the satellite? Round your answer to the nearest whole number.

31. Think About a Plan The gazebo in the photo is built in the shape of a regular octagon. Each side is 8 ft long, and the enclosed area is 310.4 ft2. What is the length of the apothem?

• How can you draw a diagram to help you solve the problem? • How can you use the area of a regular polygon formula?

32. A regular hexagon has perimeter 120 m. Find its area.

33. The area of a regular polygon is 36 in.2. Find the length of a side if the polygon has the given number of sides. Round your answer to the nearest tenth.

a. 3 b. 4 c. 6 d. Estimation Suppose the polygon is a pentagon. What would

you expect the length of a side to be? Explain.

34. A portion of a regular decagon has radii and an apothem drawn. Find the measure of each numbered angle.

35. Writing Explain why the radius of a regular polygon is greater than the apothem.

36. Constructions Use a compass to construct a circle. a. Construct two perpendicular diameters of the circle. b. Construct diameters that bisect each of the four right angles. c. Connect the consecutive points where the diameters intersect the circle. What

regular polygon have you constructed? d. Reasoning How can a circle help you construct a regular hexagon?

Find the perimeter and area of each regular polygon. Round to the nearest tenth, as necessary.

37. a square with vertices at (-1, 0), (2, 3), (5, 0) and (2, -3)

38. an equilateral triangle with two vertices at (-4, 1) and (4, 7)

39. a hexagon with two adjacent vertices at (-2, 1) and (1, 2)

40. To find the area of an equilateral triangle, you can use the formula A = 1

2bh or A = 12ap. A third way to find the area of

an equilateral triangle is to use the formula A = 14s213.

Verify the formula A = 14s213 in two ways as follows:

a. Find the area of Figure 1 using the formula A = 12bh.

b. Find the area of Figure 2 using the formula A = 12ap.

41. For Problem 1 on page 629, write a proof that the apothem bisects the vertex angle of an isosceles triangle formed by two radii.


32

1


Figure 1

s

s2

s

s2

Figure 2

Proof

STEM

634 Chapter 10 Area

42. Prove that the bisectors of the angles of a regular polygon are concurrent and that they are, in fact, radii of the polygon. (Hint: For regular n-gon ABCDE . . ., let P be the intersection of the bisectors of ∠ABC and ∠BCD. Show that DP must be the bisector of ∠CDE.)

43. Coordinate Geometry A regular octagon with center at the origin and radius 4 is graphed in the coordinate plane.

a. Since V2 lies on the line y = x, its x- and y-coordinates are equal. Use the Distance Formula to find the coordinates of V2 to the nearest tenth.

b. Use the coordinates of V2 and the formula A = 12bh to find

the area of △V1OV2 to the nearest tenth. c. Use your answer to part (b) to find the area of the octagon to

the nearest whole number.

ChallengeCProof


�2 2O

2

y

x

�2

V2

V1 (4, 0)

Mixed Review

48. What is the area of a kite with diagonals 8 m and 11.5 m?

49. The area of a trapezoid is 42 m2. The trapezoid has a height of 7 m and one base of 4 m. What is the length of the other base?

Get Ready! To prepare for Lesson 10-4, do Exercises 50–52.

Find the perimeter and area of each figure.

50. 51. 52.

See Lesson 10-2.

See Lesson 1-8.


7 in.


8 m

4 m


8 cm

6 cm

Standardized Test Prep

44. What is the area of a regular pentagon with an apothem of 25.1 mm and perimeter of 182 mm?

913.6 mm2 2284.1 mm2 3654.6 mm2 4568.2 mm2

45. What is the most precise name for a regular polygon with four right angles?

square parallelogram trapezoid rectangle

46. △ABC has coordinates A(-2, 4), B(3, 1), and C(0, -2). If you reflect △ABC across the x-axis, what are the coordinates of the vertices of the image △A′B′C′?

A′(2, 4), B′(-3, 1), C′(0, -2) A′(4, -2), B′(1, 3), C′(-2, 0)

A′(-2, -4), B′(3, -1), C′(0, 2) A′(4, 2), B′(1, -3), C′(-2, 0)

47. An equilateral triangle on a coordinate grid has vertices at (0, 0) and (4, 0). What are the possible locations of the third vertex?

SAT/ACT

ShortResponse

Documents

Objective - Mrs. Meyer's Math SiteProblem 1 Lesson 10-3 Areas of Regular Polygons 629 Objective To find the area of a regular polygon 10-3 Areas of Regular Polygons You want to build