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11.3 Areas of Polygons

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11.3 Areas of Polygons 11.3 Areas of Polygons EEssentialssential QQuestionuestion How can you fi nd the area of a regular polygon? CP The center of a regular polygon is the center P apothem of its circumscribed circle. 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 geometry software to construct each regular polygon with side lengths 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 pentagon 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 33:16:16 PMPM 11.3 Lesson WWhathat YYouou WWillill LLearnearn Find areas of rhombuses and kites. Find angle measures in regular polygons. Core VocabularyVocabulary Find areas of regular polygons. center of a regular polygon, p. 611 Finding Areas of Rhombuses and Kites radius of a regular polygon, d d p. 611 You can divide a rhombus or kite with diagonals 1 and 2 into two congruent triangles 1 1 1 1 apothem of a regular polygon, with base 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 square meters. So, the area is 35 square centimeters. MMonitoringonitoring PProgressrogress 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 Circumference, Area, and Volume hhs_geo_pe_1103.indds_geo_pe_1103.indd 610610 11/19/15/19/15 33:16:16 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 circumscribed circle. 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 triangle that has two radii as legs. The word “apothem” is a central angle. refers to a segment as well as a length. 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°. isosceles triangle, 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 PProgressrogress 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 33:16:16 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 point 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 perimeter 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 nonagon 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 ratios 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 33:16:16 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 octagon 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 Pythagorean Theorem (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 PProgressrogress 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 33:16:16 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.
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