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Home , Apothem, Dodecagon, Hexagon, Kite (geometry), Nonagon, Polygon

11.3 of

EEssentialssential QQuestionuestion How can you fi nd the of a regular ? CP The center of a is the center P of its circumscribed . The distance from the center to any side of a regular polygon is called the apothem of a C center regular polygon.

Finding the Area of a Regular Polygon Work with a partner. Use dynamic software to construct each regular polygon with side of 4, as shown. Find the apothem and use it to fi nd the area of the polygon. Describe the steps that you used.

a. b. 7

4 D 6 C 5 3

E 4 C

2 3

2 1

1 A 0 B A 0 B −3 −2 −1 031 2 −5 −4 −3 −2 −1 03451 2

c. 8 d. F 10 E E D 9 7 8 6 G 7 D

5 6

4 5 F C 4 3 H 3 C 2 2

1 1

A 0 B A 0 B −5 −4 −3 −2 −1 03451 2 −6 −5 −4 −3 −2 −1 01 2 3456

Writing a Formula for Area Work with a partner. Generalize the steps you used in Exploration 1 to develop a REASONING formula for the area of a regular polygon. ABSTRACTLY To be profi cient in math, CCommunicateommunicate YourYour AnswerAnswer you need to know and fl exibly use different 3. How can you fi nd the area of a regular polygon? properties of operations 4. Regular ABCDE has side lengths of 6 meters and an apothem of and objects. approximately 4.13 meters. Find the area of ABCDE.

Section 11.3 Areas of Polygons 609

hhs_geo_pe_1103.indds_geo_pe_1103.indd 609609 11/19/15/19/15 3:163:16 PMPM 11.3 Lesson WWhathat YYouou WWillill LLearnearn Find areas of and kites. Find measures in regular polygons. Core VocabularyVocabulary Find areas of regular polygons. center of a regular polygon, p. 611 Finding Areas of Rhombuses and Kites of a regular polygon, d d p. 611 You can divide a or with 1 and 2 into two congruent 1 1 1 1 apothem of a regular polygon, with d , height — d , and area — d — d = — d d . So, the area of a rhombus or kite 1 2 2 2 1( 2 2) 4 1 2 p. 611 —1 d d = —1 d d is 2 ( 1 2 ) 1 2. central angle of a regular 4 2 polygon, p. 611 A = 1 d d A = 1 d d 4 1 2 4 1 2 Previous 1 d 1 d d 2 2 d 2 2 rhombus 2 2 kite

d d 1 1 CCoreore CConceptoncept Area of a Rhombus or Kite 1 d d — d d The area of a rhombus or kite with diagonals 1 and 2 is 2 1 2.

d d 2 2

d d 1 1

Finding the Area of a Rhombus or Kite

Find the area of each rhombus or kite. a. b.

8 m 7 cm

6 m 10 cm

SOLUTION

1 1 A = — d d A = — d d a. 2 1 2 b. 2 1 2 1 1 = — = — 2 (6)(8) 2 (10)(7) = 24 = 35

So, the area is 24 meters. So, the area is 35 square centimeters.

MMonitoringonitoring ProgressProgress Help in English and Spanish at BigIdeasMath.com d = d = 1. Find the area of a rhombus with diagonals 1 4 feet and 2 5 feet. d = d = 2. Find the area of a kite with diagonals 1 12 inches and 2 9 inches.

610 Chapter 11 , Area, and Volume

hhs_geo_pe_1103.indds_geo_pe_1103.indd 610610 11/19/15/19/15 3:163:16 PMPM Finding Angle Measures in Regular Polygons The diagram shows a regular polygon inscribed in a circle. M The center of a regular polygon and the radius of a apothem regular polygon are the center and the radius of its Q PQ center . P N The distance from the center to any side of a regular radius PN polygon is called the apothem of a regular polygon. The apothem is the height to the base of an isosceles ∠MPN that has two radii as legs. The word “apothem” is a central angle. refers to a segment as well as a . For a given regular polygon, think of an apothem as a segment and the apothem as a length. A central angle of a regular polygon is an angle formed by two radii drawn to consecutive vertices of the polygon. To fi nd the measure of each central angle, divide 360° by the number of sides.

Finding Angle Measures in a Regular Polygon

In the diagram, ABCDE is a regular pentagon inscribed in ⊙F. Find each angle measure. C

B D F G

A E

m∠AFB m∠AFG m∠GAF ANALYZING a. b. c. RELATIONSHIPS SOLUTION FG — is an altitude of an 360° a. ∠AFB is a central angle, so m∠AFB = — = 72°. , so 5 it is also a median and — angle bisector of the b. FG is an apothem, which makes it an altitude of isosceles △AFB. isosceles triangle. — 1 So, FG bisects ∠AFB and m∠AFG = — m∠AFB = 36°. 2 c. By the Triangle Sum Theorem (Theorem 5.1), the sum of the angle measures of right △GAF is 180°. m∠GAF = 180° − 90° − 36° = 54° So, m∠GAF = 54°.

MMonitoringonitoring ProgressProgress Help in English and Spanish at BigIdeasMath.com

In the diagram, WXYZ is a square inscribed in ⊙P. X Y Q 3. Identify the center, a radius, an apothem, and a central angle of the polygon. P 4. Find m∠XPY, m∠XPQ, and m∠PXQ. W Z

Section 11.3 Areas of Polygons 611

hhs_geo_pe_1103.indds_geo_pe_1103.indd 611611 11/19/15/19/15 3:163:16 PMPM Finding Areas of Regular Polygons You can fi nd the area of any regular n-gon by dividing it into congruent triangles. A = Area of one triangle ⋅ Number of triangles

READING 1 Base of triangle is s and height of = — ⋅ s ⋅ a ⋅ n DIAGRAMS ( 2 ) triangle is a. Number of triangles is n.

In this book, a 1 Commutative and Associative = — a n s 2 ⋅ ⋅ ( ⋅ ) shown inside a regular Properties of Multiplication a s polygon marks the center 1 There are n congruent sides of of the circle that can be = — a ⋅ P 2 length s, so P is n ⋅ s. circumscribed about the polygon. CCoreore CConceptoncept Area of a Regular Polygon The area of a regular n-gon with side length s is one-half the product of the apothem a and the perimeter P.

1 1 a A = — aP A = — a ns s 2 , or 2 ⋅

Finding the Area of a Regular Polygon

A regular is inscribed in a circle with a radius of 4 units. Find the area of the nonagon.

K 4 L M 4 J

SOLUTION 360° — ∠JLK — ° LM The measure of central is 9 , or 40 . Apothem bisects the central angle, so m∠KLM is 20°. To fi nd the lengths of the legs, use trigonometric for right △KLM. L 20° 4 4

J M K MK LM ° = — ° = — sin 20 LK cos 20 LK MK LM sin 20° = — cos 20° = — 4 4 4 sin 20° = MK 4 cos 20° = LM The regular nonagon has side length s = 2(MK) = 2(4 sin 20°) = 8 sin 20°, and apothem a = LM = 4 cos 20°.

1 1 A = — a ns = — ° ° ≈ So, the area is 2 ⋅ 2 (4 cos 20 ) ⋅ (9)(8 sin 20 ) 46.3 square units.

612 Chapter 11 Circumference, Area, and Volume

hhs_geo_pe_1103.indds_geo_pe_1103.indd 612612 11/19/15/19/15 3:163:16 PMPM Finding the Area of a Regular Polygon

You are decorating the top of a table by covering it with small ceramic tiles. The tabletop is a regular with 15-inch sides and a radius of about 19.6 inches. What is the area you are covering?

R 15 in. 19.6 in.

PQ

SOLUTION Step 1 Find the perimeter P of the tabletop. An octagon has 8 sides, so P = 8(15) = 120 inches. Step 2 Find the apothem a. The apothem is height RS of △PQR. — — Because △PQR is isosceles, altitude RS bisects QP . 1 1 QS = — QP = — = So, 2 ( ) 2 (15) 7.5 inches. R

19.6 in.

P S Q 7.5 in.

To fi nd RS, use the (Theorem 9.1) for △RQS. —— — a = RS = √ 19.62 − 7.52 = √327.91 ≈ 18.108 Step 3 Find the area A of the tabletop.

1 A = — aP 2 Formula for area of a regular polygon — 1 = — √ 2 ( 327.91 ) (120) Substitute. ≈ 1086.5 Simplify.

The area you are covering with tiles is about 1086.5 square inches.

MMonitoringonitoring ProgressProgress Help in English and Spanish at BigIdeasMath.com Find the area of the regular polygon.

5. 6. 7

8 6.5

Section 11.3 Areas of Polygons 613

hhs_geo_pe_1103.indds_geo_pe_1103.indd 613613 11/19/15/19/15 3:163:16 PMPM 11.3 Exercises Dynamic Solutions available at BigIdeasMath.com

VVocabularyocabulary andand CoreCore ConceptConcept CheckCheck 1. WRITING Explain how to fi nd the measure of a central angle of a regular polygon.

2. DIFFERENT WORDS, SAME QUESTION Which is different? Find “both” answers. B A 8 Find the radius of ⊙F. Find the apothem of polygon ABCDE. 5.5 G C F 6.8 Find AF. Find the radius of polygon ABCDE. E D

MMonitoringonitoring ProgressProgress andand ModelingModeling withwith MathematicsMathematics

In Exercises 3–6, fi nd the area of the kite or rhombus. In Exercises 15–18, fi nd the given A B (See Example 1.) angle measure for regular octagon ABCDEFGH. (See Example 2.) H C 3. 4. J K 19 6 15. m∠GJH 16. m∠GJK G D 210 38 17. m∠KGJ 18. m∠EJH 6 F E

5. 6. In Exercises 19–24, fi nd the area of the regular polygon. 5 (See Examples 3 and 4.) 5 7 7 6 19. 20. 5 12 6.84 10 2 3 In Exercises 7–10, use the diagram. J 7. Identify the center of N 5.88 polygon JKLMN. 21. 22. 4.05 7 Q K P 8. Identify a central angle 2.77 JKLMN. 5 of polygon M 2.5 L 9. What is the radius of polygon JKLMN? 23. an octagon with a radius of 11 units 10. What is the apothem of polygon JKLMN? 24. a pentagon with an apothem of 5 units In Exercises 11–14, fi nd the measure of a central angle 25. ERROR ANALYSIS Describe and correct the error in of a regular polygon with the given number of sides. fi nding the area of the kite. Round answers to the nearest tenth of a , if necessary. 1 3.6 5.4 A = — (3.6)(5.4) 11. 10 sides 12. 18 sides ✗ 2 2 = 3 2 5 9.72 13. 24 sides 14. 7 sides So, the area of the kite is 9.72 square units.

614 Chapter 11 Circumference, Area, and Volume

hhs_geo_pe_1103.indds_geo_pe_1103.indd 614614 11/19/15/19/15 3:163:16 PMPM 26. ERROR ANALYSIS Describe and correct the error in CRITICAL THINKING In Exercises 33–35, tell whether the fi nding the area of the regular . statement is true or false. Explain your reasoning.

— 33. The area of a regular n-gon of a fi xed radius r s = √ 2 − 2 ≈ ✗ 15 13 7.5 increases as n increases. 1 A = — a ns 2 ⋅ 15 1 34. The apothem of a regular polygon is always less than ≈ — (13)(6)(7.5) 13 2 the radius. = 292.5 So, the area of the hexagon is about 35. The radius of a regular polygon is always less than the 292.5 square units. side length. 36. REASONING Predict which fi gure has the greatest In Exercises 27–30, fi nd the area of the shaded region. area and which has the least area. Explain your reasoning. Check by fi nding the area of each figure. 27. 28. ○A ○B 14 13 in. 12

9 in.

29. 30. C 2 3 ○ 15 in. 8 60°

18 in.

31. MODELING WITH MATHEMATICS Basaltic columns 37. USING EQUATIONS Find the area of a regular are geological formations that result from rapidly pentagon inscribed in a circle whose equation is given x − 2 + y + 2 = cooling lava. Giant’s Causeway in Ireland contains by ( 4) ( 2) 25. many hexagonal basaltic columns. Suppose the top of one of the columns is in the of a regular 38. REASONING What happens to the area of a kite if hexagon with a radius of 8 inches. Find the area of the you double the length of one of the diagonals? top of the column to the nearest square inch. if you double the length of both diagonals? Justify your answer.

MATHEMATICAL CONNECTIONS In Exercises 39 and 40, write and solve an equation to fi nd the indicated lengths. Round decimal answers to the nearest tenth.

39. The area of a kite is 324 square inches. One is twice as long as the other diagonal. Find the length of each diagonal.

40. One diagonal of a rhombus is four times the length 32. MODELING WITH MATHEMATICS A watch has a of the other diagonal. The area of the rhombus is circular surface on a background that is a regular 98 square feet. Find the length of each diagonal. octagon. Find the area of the octagon. Then fi nd the area of the silver border around the circular . 41. REASONING The perimeter of a regular nonagon, 1 cm or 9-gon, is 18 inches. Is this enough information to 0.2 cm fi nd the area? If so, fi nd the area and explain your reasoning. If not, explain why not.

Section 11.3 Areas of Polygons 615

hs_geo_pe_1103.indd 615 3/9/16 9:45 AM 42. MAKING AN ARGUMENT Your friend claims that it is 49. USING STRUCTURE In the fi gure, an equilateral possible to fi nd the area of any rhombus if you only triangle lies inside a square inside a regular pentagon know the perimeter of the rhombus. Is your friend inside a regular hexagon. Find the approximate area of correct? Explain your reasoning. the entire shaded region to the nearest whole number.

43. PROOF Prove that the area of any with 1 A = — d d d d diagonals is 2 1 2, where 1 and 2 are the lengths of the diagonals. Q 8 PR T d 1 50. THOUGHT PROVOKING The area of a regular n-gon is 1 S given by A = — aP. As n approaches infi nity, what does d 2 2 the n-gon approach? What does P approach? What does a approach? What can you conclude from your three answers? Explain your reasoning. 44. HOW DO YOU SEE IT? Explain how to fi nd the area of the regular hexagon by dividing the hexagon into equilateral triangles. U V

Z W

Y X 51. COMPARING METHODS Find the area of regular 1 ABCDE A = — aP pentagon by using the formula 2 , or 1 A = — a • ns 2 . Then fi nd the area by adding the areas 45. REWRITING A FORMULA Rewrite the formula for of smaller polygons. Check that both methods yield the area of a rhombus for the special case of a square the same area. Which method do you prefer? Explain with side length s. Show that this is the same as the your reasoning. formula for the area of a square, A = s2. A 5 46. REWRITING A FORMULA Use the formula for the E B area of a regular polygon to show that the area of an P

equilateral— triangle can be found by using the formula 1 A= — s2√ s D C 4 3 , where is the side length.

47. CRITICAL THINKING The area of a regular pentagon 52. USING STRUCTURE Two regular polygons both have is 72 square centimeters. Find the length of one side. n sides. One of the polygons is inscribed in, and the other is circumscribed about, a circle of radius r. Find 48. CRITICAL THINKING The area of a , or the area between the two polygons in terms of n and r. 12-gon, is 140 square inches. Find the apothem of the polygon.

MMaintainingaintaining MathematicalMathematical ProficiencyProficiency Reviewing what you learned in previous grades and lessons Determine whether the fi gure has , , both, or neither. If the fi gure has line symmetry, determine the number of lines of symmetry. If the fi gure has rotational symmetry, describe any rotations that map the fi gure onto itself. (Section 4.2 and Section 4.3) 53. 54. 55. 56.

616 Chapter 11 Circumference, Area, and Volume

hhs_geo_pe_1103.indds_geo_pe_1103.indd 616616 11/19/15/19/15 3:173:17 PMPM