Areas of Regular Polygons. To find the area of a regular polygon. The Solve It involves the area of a polygon.

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1 10- Area of Regular Polygon Common Core State Standard G-MG.A.1 Ue geometric hape, their meaure, and their propertie to decribe object. Alo G-CO.D.1 MP 1, MP, MP 4, MP 6, MP 7 Objective To find the area of a regular polygon Solve a impler problem. Try uing fewer ide to ee what happen. You want to build a koi pond. For the border, you plan to ue -ft-long piece of wood. You have 1 piece that you can connect together at any angle, including a traight angle. If you want to maximize the area of the pond, in what hape hould you arrange the piece? Explain your reaoning. MATHEMATICAL PRACTICES The Solve It involve the area of a polygon. Leon L VocabularyV radiu of a regular polygon apothem Eential Undertanding The area of a regular polygon i related to the ditance from the center to a ide. You can circumcribe a circle about any regular polygon. The center of a regular polygon i the center of the circumcribed circle. The radiu of a regular polygon i the ditance from the center to a vertex. The apothem i the perpendicular ditance from the center to a ide. Center Radiu Apothem How do you know the radii make iocele triangle? Since the pentagon i a regular polygon, the radii are congruent. So, the triangle made by two adjacent radii and a ide of the polygon i an iocele triangle. Problem 1 Finding Angle Meaure The figure at the right i a regular pentagon with radii and an apothem drawn. What i the meaure of each numbered angle? m 1 = 60 5 = 7 Divide 60 by the number of ide. 1 m = 1 m 1 The apothem biect the vertex angle of the iocele triangle formed by the radii. = 1 (7) = m = 180 The um of the meaure of the angle of a triangle i 180. m = 54 m 1 = 7, m = 6, and m = 54. Got It? 1. At the right, a portion of a regular octagon ha radii and an apothem drawn. What i the meaure of each numbered angle? 1 Leon 10- Area of Regular Polygon 69

2 Potulate 10-1 If two figure are congruent, then their area are equal. Suppoe you have a regular n-gon with ide. The radii divide the figure into n congruent iocele triangle. By Potulate 10-1, the area of the iocele triangle are equal. Each triangle ha a height of a and a bae of length, o the area of each triangle i 1 a. Since there are n congruent triangle, the area of the n-gon i A = n # 1 a. The perimeter p of the n-gon i the number of ide n time the length of a ide, or n. By ubtitution, the area can be expreed a A = 1 ap. a Theorem 10-6 Area of a Regular Polygon The area of a regular polygon i half the product of the apothem and the perimeter. A = 1 ap p a What do you know about the regular decagon? A decagon ha 10 ide, o n = 10. From the diagram, you know that the apothem a i 1. in., and the ide length i 8 in. Problem Finding the Area of a Regular Polygon What i the area of the regular decagon at the right? Step 1 Find the perimeter of the regular decagon. p = n Ue the formula for the perimeter of an n-gon. = 10(8) Subtitute 10 for n and 8 for. = 80 in. Step A = 1 ap Find the area of the regular decagon. Ue the formula for the area of a regular polygon. = 1 (1.)(80) Subtitute 1. for a and 80 for p. = 49 The regular decagon ha an area of 49 in.. 8 in. 1. in. Got It?. a. What i the area of a regular pentagon with an 8-cm apothem and 11.6-cm ide? b. Reaoning If the ide of a regular polygon i reduced to half it length, how doe the perimeter of the polygon change? Explain. 60 Chapter 10 Area

3 Problem Uing Special Triangle to Find Area STEM Zoology A honeycomb i made up of regular hexagonal cell. The length of a ide of a cell i mm. What i the area of a cell? You know the length of a ide, which you can ue to find the perimeter. The apothem Draw a diagram to help find the apothem. Then ue the area formula for a regular polygon. Step 1 Find the apothem. The radii form ix 60 angle at the center, o you can ue a triangle to find the apothem. Step a = 1.51 longer leg = 1 # horter leg Find the perimeter. p = n Ue the formula for the perimeter of an n-gon. Step = 6() Subtitute 6 for n and for. = 18 mm A = 1 ap Find the area. Ue the formula for the area of a regular polygon. = 1 (1.51) (18) Subtitute 1.51 for a and 18 for p The area i about mm. Ue a calculator a 1.5 mm mm Got It?. The ide of a regular hexagon i 16 ft. What i the area of the hexagon? Round your anwer to the nearet quare foot. Leon Check Do you know HOW? What i the area of each regular polygon? Round your anwer to the nearet tenth in. ft Do you UNDERSTAND? MATHEMATICAL PRACTICES 5. Vocabulary What i the difference between a radiu and an apothem? 6. What i the relationhip between the ide length and the apothem in each figure? a. a quare b. a regular hexagon c. an equilateral triangle. 4. m 4 V 7. Error Analyi Your friend ay you can ue pecial triangle to find the apothem of any regular polygon. What i your friend error? Explain. Leon 10- Area of Regular Polygon 61

4 Practice and Problem-Solving Exercie MATHEMATICAL PRACTICES A Practice Each regular polygon ha radii and apothem a hown. Find the meaure of each numbered angle. See Problem Find the area of each regular polygon with the given apothem a and ide length. See Problem. 11. pentagon, a = 4. cm, = 5. cm 1. 7-gon, a = 9.1 ft, = 8 ft 1. octagon, a = 60.4 in., = 50 in. 14. nonagon, a = 7.5 in., = 0 in. 15. decagon, a = 19 m, = 1. m 16. dodecagon, a = 6.1 cm, = 14 cm Find the area of each regular polygon. Round your anwer to the nearet tenth ft in. 6 m See Problem. 0. Art You are painting a mural of colored equilateral triangle. The radiu of each triangle i 1.7 in. What i the area of each triangle to the nearet quare inch? 0 Find the area of each regular polygon with the given radiu or apothem. If your anwer i not an integer, leave it in implet radical form in. 6 cm 8V in in. 6V m5 m B Apply Find the meaure of the angle formed by (a) two conecutive radii and (b) a radiu and a ide of the given regular polygon. 6. pentagon 7. octagon 8. nonagon 9. dodecagon 6 Chapter 10 Area

5 STEM 0. Satellite One of the mallet pace atellite ever developed ha the hape of a pyramid. Each of the four face of the pyramid i an equilateral triangle with ide about 1 cm long. What i the area of one equilateral triangular face of the atellite? Round your anwer to the nearet whole number. 1. Think About a Plan The gazebo in the photo i built in the hape of a regular octagon. Each ide i 8 ft long, and the encloed area i 10.4 ft. What i the length of the apothem? How can you draw a diagram to help you olve the problem? How can you ue the area of a regular polygon formula?. A regular hexagon ha perimeter 10 m. Find it area.. The area of a regular polygon i 6 in.. Find the length of a ide if the polygon ha the given number of ide. Round your anwer to the nearet tenth. a. b. 4 c. 6 d. Etimation Suppoe the polygon i a pentagon. What would you expect the length of a ide to be? Explain. 4. A portion of a regular decagon ha radii and an apothem drawn. Find the meaure of each numbered angle. 5. Writing Explain why the radiu of a regular polygon i greater than the apothem. 6. Contruction Ue a compa to contruct a circle. a. Contruct two perpendicular diameter of the circle. b. Contruct diameter that biect each of the four right angle. c. Connect the conecutive point where the diameter interect the circle. What regular polygon have you contructed? d. Reaoning How can a circle help you contruct a regular hexagon? Find the perimeter and area of each regular polygon. Round to the nearet tenth, a neceary. 7. a quare with vertice at (-1, 0), (, ), (5, 0) and (, -) 8. an equilateral triangle with two vertice at (-4, 1) and (4, 7) 9. a hexagon with two adjacent vertice at (-, 1) and (1, ) To find the area of an equilateral triangle, you can ue the formula A = 1 bh or A = 1 ap. A third way to find the area of an equilateral triangle i to ue the formula A = Verify the formula A = in two way a follow: a. Find the area of Figure 1 uing the formula A = 1 bh. b. Find the area of Figure uing the formula A = 1 ap. 41. For Problem 1 on page 69, write a proof that the apothem Proof biect the vertex angle of an iocele triangle formed by two radii. Figure 1 Figure Leon 10- Area of Regular Polygon 6

6 C Challenge 4. Prove that the biector of the angle of a regular polygon are concurrent and that Proof they are, in fact, radii of the polygon. (Hint: For regular n-gon ABCDE..., let P be the interection of the biector of ABC and BCD. Show that DP mut be the biector of CDE.) 4. Coordinate Geometry A regular octagon with center at the origin and radiu 4 i graphed in the coordinate plane. a. Since V lie on the line y = x, it x- and y-coordinate are equal. Ue the Ditance Formula to find the coordinate of V to the nearet tenth. b. Ue the coordinate of V and the formula A = 1 bh to find the area of V 1 OV to the nearet tenth. c. Ue your anwer to part (b) to find the area of the octagon to the nearet whole number. y O V V 1 (4, 0) x Standardized Tet Prep SAT/ACT 44. What i the area of a regular pentagon with an apothem of 5.1 mm and perimeter of 18 mm? 91.6 mm 84.1 mm mm mm 45. What i the mot precie name for a regular polygon with four right angle? quare parallelogram trapezoid rectangle 46. ABC ha coordinate A(-, 4), B(, 1), and C(0, -). If you reflect ABC acro the x-axi, what are the coordinate of the vertice of the image A B C? A (, 4), B (-, 1), C (0, -) A (4, -), B (1, ), C (-, 0) A (-, -4), B (, -1), C (0, ) A (4, ), B (1, -), C (-, 0) Short Repone 47. An equilateral triangle on a coordinate grid ha vertice at (0, 0) and (4, 0). What are the poible location of the third vertex? Mixed Review 48. What i the area of a kite with diagonal 8 m and 11.5 m? 49. The area of a trapezoid i 4 m. The trapezoid ha a height of 7 m and one bae of 4 m. What i the length of the other bae? See Leon 10-. Get Ready! To prepare for Leon 10-4, do Exercie Find the perimeter and area of each figure m See Leon cm 7 in. 8 m 8 cm 64 Chapter 10 Area