03 Areas of RegularPolygons

ObjecHve To find the area of a regular polygon

Mathematics Florida StandardsMAFS.912.G-MG.1.1 Use geometric shapes, theirmeasures, and their properties to describe objects.Also MAFS.912.G-C0.4.13

MP1, MP3, MP 4, MP6. MP 7

f/^ iGe^ngReod^ X C 4

Solve a simplerproblem. Try usingfewer sides to see

what happens.

You want to build a koi pond. For the border, you plan to use 3-ft-longpieces of wood. You have 12 pieces that you can connect together atany angle, including a straight angle. If you want to maximize the areaof the pond, in what shape should you arrange the pieces? Exploin yourreasoning.

MATHEMATICAL

PRACTICES

Lesson

Vocabulary• radius of a

regular polygonapothem

The Solve It involves the area of a polygon.

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

You can circumscribe a circle about any regular polygon. Thecenter of a regular polygon is the center of the circumscribed

circle. The radius of a regular polygon is the distance from thecenter to a vertex. The apothem is the perpendicular distancefrom the center to a side.

Center

Radius

Apothem

How do you knowthe radii make

isosceles triangles?Since the pentagon isa regular polygon, theradii are congruent. So,the triangle made bytwo adjacent radii and aside of the polygon is anisosceles triangle.

Problem 1 Finding Angle Measures

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

mZl = 5|° = 72

m/.2 = |m/ll

= i(72) = 36

Divide 360 by the number of sides,

The apothem bisects the vertex angle of theisosceles triangle formed by the radii.

90 + 36 + mZ.3 = 180 The sum of the measures of the angles of a triangle Is 180.

m/L3 = 54

m/.l = 72, m/-2 = 36, and mZ.2 = 54.

Got It? 1. At the right, a portion of a regular octagon has radiiand an apothem drawn. What is the measure of each

numbered angle?

PowerGeometry.com { Lesson 10-3 Areas of Regular Polygons 629

Postulate 10-1

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

Suppose you have a regular «-gon with side s. The radii divide the figureinto n congruent isosceles triangles. By Postulate 10-1, the areas of theisosceles triangles are equal. Each triangle has a height of a and a base

of length s, so the area of each triangle is ̂ as.

Since there are n congruent triangles, the area of the n-gon is

A = n ' \as. The perimeter p of the n-gon is the number of sides n timesthe length of a side s, or ns. By substitution, the area can be expressed asA = ̂ap.

\

V

/r\ '

/ I \

/^1 \'ri

Theorem 10-6 Area of a Regular Polygon

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

A = k.2^P

J

Pi

What do you knowabout the regulardecagon?A decagon has 10 sides,so n = 10. From the

diagram, you knowthat the apothem a is12.3 in., and the side

length sis 8 in.

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 fern and 8fors.

= 80 in.

Step 2 Find the area of the regular decagon.

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

= |(12.3)(80) Substitute 12.3 for a and 80 for p.= 492

The regular decagon has an area of 492 In.^.

Got It? 2. a. What is the area of a regular pentagon with an 8-cm apothem and11.6-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.

i12.3 in

8 in.

630 Chapter 10 Area

Problem 3 Using Special Triangles to Find Area @21!)

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

You know the length of aside, which you can use tofind the perimeter.

The apothem Draw a diagram to help find theapothem. Then use the area formulafor a regular polygon.

&

30"/ la'^ /3mm'ILL[—11.5 mm

Step 1 Find the apothem.

The radii form six 60° angles at the center, so you can use a30°-60°-9D'' triangle to find the apothem.

a = 1.5 V3 longer leg = V3 • 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 = ̂ap Use the formula for the area of a regular polygon.

= |(1.5 V^) (18) Substitute 1.5 V3 for a and 18 for p.= 23.3826859 Use a calculator.

The area is about 23 mm^.

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

Lesson Check

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

1.

n5 in.

MATHEMATICAL

Do you UNDERSTAND? PRACTICES

5. Vocabulary What is the difference between a radiusand an apothem?

6. What is the relationship between the side length andthe apothem in each figure?a. a square

b. a regular hexagonc. an equilateral triangle

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

C PowerGeometry.com Lesson 10-3 Areas of Regular Polygons 631

Practice and Problem-Solving ExercisesMATHEMATICAL

PRACTICES

Practice Each regular polygon has radii and apothem as shown. Find the measure

of each numbered angle.

^ See Problem 1.

10.

Find the area of each regular polygon with the given apothem a and side ^ See Problem 2.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, ft = 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. See Problem 3.18. A 19.17.

18ft

6 m

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

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

8"v/3 in.

^ Apply

23.

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

4 in.

29. dodecagon

632 Chapter 10 Area

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 sidesabout 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 ft^. Wliat 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.^. Find the length of a side ifthe 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

apotliem.

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)

I

40. To find the area of an equilateral triangle, you can use the

formula A = ̂bh or A = ̂ap. A third way to find the area ofan equilateral triangle is to use the formula A =

Verify the formula A = in two ways as follows:

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

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

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

Figure 1 Figure 2

c PowerGeometry.com Lesson 10-3 Areas of Regular Polygons 633

Challenge Prove that the bisectors of the angles of a regular polygon are concurrent and thatNil# Proof ̂ hey are, in fact, radii of the polygon. {Hint: For regular n-gon ABCDE..., let P be

the intersection of the bisectors of /LABC and /LBCD. Show that 'DP must be the

bisector of /LCDE.)

43. Coordinate Geometry A regular octagon with center at theorigin and radius 4 is graphed in the coordinate plane.a. Since V2 lies on the line y = x, its x- andy-coordinates are

equal. Use the Distance Formula to find the coordinates of V2to the nearest tenth.

b. Use the coordinates of V2 and the formula A = ̂bh to findthe area of A V]0V2 to the nearest tenth.

c. Use your answer to part (b) to find the area of the octagon tothe nearest whole number.

y

/ \-/ %

1 1/. .

V"

\1

(4 0)*

—

V-2_M \ -2—/\ ' /

X

—

|\ \i

~"rri'

r./

Standardized Test Prep

SAT/Aa

Short

. Response

44. What is the area of a regular pentagon with an apothem of 25. i mm and perimeter

of 182 mm?

CA^ 913.6 mm^ CX^ 2284.1 mm^ CO 3654.6 mm^ CO 4568.2 mm^

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

CO square CO parallelogram CO trapezoid CQ rectangle

46. AABC has coordinates yl(-2,4), B(3,1), and C(0, -2). If you reflect AABC across

the x-axis, what are the coordinates of the vertices of the image AA 'B'C'l

CO A'{2, 4), B'(-3, 1), C'CO, -2) CO ̂'(4, -2), B'(l. 3), C'(-2, 0)

CO A'{-2, -4). B'{3, -1), C'CO, 2) CO 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 arethe possible locations of the third vertex?

rMixed Review

48. What is the area of a kite with diagonals 8 m and 11.5 m? ^ See Lesson 10-2.

49. The area of a trapezoid is 42 m^. 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. n-. H 52.cr

--4 m

7 in. 8 m

^ See Lesson 1-8.

6 cm

8 cm

634 Chapter 10 Area