11.6 Areas of Regular Polygons

Transcription

1 11.6 Areas of egular Polygons Goal p Find areas of regular polygons inscribed in circles. Your Notes VOCABUAY Center of a polygon adius of a polygon Apothem of a polygon Central angle of a regular polygon Example 1 Find angle measures in a regular polygon In the diagram, ABCDEF is a regular C D hexagon inscribed in (G. Find each angle measure. a. m EGF H b. m EGH A F c. m HEG Solution B G E a. EGF is a central angle, so m EGF, or. b. } GH is an apothem, which makes it an of isosceles negf. So, } GH EGF and m EGH m EGF. c. The sum of the measures of right nheg is So, 1 1 m HEG 1808, and m HEG. 316 esson 11.6 Geometry Notetaking Guide Copyright Holt McDougal. All rights reserved.

2 11.6 Areas of egular Polygons Goal p Find areas of regular polygons inscribed in circles. Your Notes VOCABUAY Center of a polygon The center of a polygon is the center of its circumscribed circle. adius of a polygon The radius of a polygon is the radius of its circumscribed circle. Apothem of a polygon The distance from the center to any side of the polygon is the apothem. Central angle of a regular polygon A central angle of a regular polygon is an angle formed by two radii drawn to consecutive vertices of the polygon. Example 1 Find angle measures in a regular polygon In the diagram, ABCDEF is a regular C D hexagon inscribed in (G. Find each angle measure. a. m EGF H b. m EGH A F c. m HEG B G E Solution a. EGF is a central angle, so m EGF } 3608, or b. } GH is an apothem, which makes it an altitude of isosceles negf. So, } GH bisects EGF and m EGH } 1 m EGF 308. c. The sum of the measures of right nheg is So, m HEG 1808, and m HEG esson 11.6 Geometry Notetaking Guide Copyright Holt McDougal. All rights reserved.

3 THEOEM 11.11: AEA OF A EGUA POYGON The area of a regular n-gon with side length s is half the product of the apothem a and the perimeter P, so A 1 }, or A 1 } p. a s Example Find the area of a regular polygon Coaster A wooden coaster is a regular octagon with 3 centimeter sides and a radius of about 3.9 centimeters. What is the area of the coaster? 3.9 cm 3 cm P Solution Step 1 Find the perimeter P of the coaster. An octagon has sides, so P ( ) centimeters. Step Find the apothem a. The apothem is height of npq. Because npq is isosceles, altitude } } S QP. So, QS (QP) ( ) cm. P S 1. cm In general, your answer will be more accurate if you avoid rounding until the last step. ound your final answers to the nearest tenth unless you are told otherwise. To find S, use the Pythagorean Theorem for nqs. a S < Ï }} Ï } < Step 3 Find the area A of the coaster. A 1 } ap Formula for area of regular polygon < } 1 ( )( ) Substitute. < Simplify. The area of the coaster is about centimeters. square Copyright Holt McDougal. All rights reserved. esson 11.6 Geometry Notetaking Guide 317

4 THEOEM 11.11: AEA OF A EGUA POYGON The area of a regular n-gon with side length s is half the product of the apothem a and the perimeter P, so a A } 1 ap, or A } 1 a p ns. s Example Find the area of a regular polygon Coaster A wooden coaster is a regular octagon with 3 centimeter sides and a radius of about 3.9 centimeters. What is the area of the coaster? 3.9 cm 3 cm P Solution Step 1 Find the perimeter P of the coaster. An octagon has 8 sides, so P 8 ( 3 ) 4 centimeters. Step Find the apothem a. The apothem is height S of npq. Because npq is isosceles, altitude } S bisects } QP. So, QS 1 } (QP) 1 } ( 3 ) 1. cm. P S 1. cm In general, your answer will be more accurate if you avoid rounding until the last step. ound your final answers to the nearest tenth unless you are told otherwise. To find S, use the Pythagorean Theorem for nqs. a S < Ï }} Ï } < 3.6 Step 3 Find the area A of the coaster. A 1 } ap Formula for area of regular polygon < } 1 ( 3.6 )( 4 ) Substitute. < 43. Simplify. The area of the coaster is about 43. square centimeters. Copyright Holt McDougal. All rights reserved. esson 11.6 Geometry Notetaking Guide 317

5 Checkpoint Complete the following exercises. 1. In the diagram, FGH is a square G H inscribed in (. Find m F and m F. F. The radius of the regular pentagon 8 in. is about 6.8 inches. Find the area to the nearest square inch. 6.8 in. T S Example 3 Find the perimeter and area of a regular polygon A regular nonagon is inscribed in a circle with radius units. Find the perimeter P and area A of the nonagon. M The measure of central is, or. Apothem } M bisects the central angle, so m M is. To find the lengths of the legs, use trigonometric ratios for right nm. sin M } cos M } M M The regular nonagon has side lengths s M ( ) M and apothem a M. So, P 9s 9( ) < units, and A } 1 ap } 1 ( )( ) < square units esson 11.6 Geometry Notetaking Guide Copyright Holt McDougal. All rights reserved.

6 Checkpoint Complete the following exercises. 1. In the diagram, FGH is a square G H inscribed in (. Find m F and m F. m F 908; m F 48 F. The radius of the regular pentagon 8 in. is about 6.8 inches. Find the area to the nearest square inch. about 110 in. 6.8 in. T S Example 3 A regular nonagon is inscribed in a circle with radius units. Find the perimeter P and area A of the nonagon. The measure of central is } 3608, or Apothem } M bisects the central angle, so m M is 08. To find the lengths of the legs, use trigonometric ratios for right nm. sin 08 M } Find the perimeter and area of a regular polygon cos 08 M } sin 08 M cos 08 M The regular nonagon has side lengths s M ( sin 08 ) 10 sin 08 M and apothem a M cos 08. So, P 9s 9( 10 sin 08 ) 90 sin 08 < 30.8 units, and A } 1 ap } 1 ( cos 08 )( 90 sin 08 ) < 7.3 square units. M esson 11.6 Geometry Notetaking Guide Copyright Holt McDougal. All rights reserved.

7 Checkpoint Complete the following exercise. 3. Find the perimeter and area of the equilateral triangle inscribed in (F. H F 6 6 G FINDING ENGTHS IN A EGUA N-GON To find the area of a regular n-gon with radius r, you may need to first find the apothem a or the side length s. You can use when you know n and as in... Pythagorean Theorem: Two measures: r and a, or r and s Example and Checkpoint Ex. 1 1 } s 1 a r Homework Special ight Triangles Any one measure: r or a or s And the value of n is 3, 4, or 6 Checkpoint Ex. 3 Trigonometry Any one measure: r or a or s Example 3 and Checkpoint Ex. 3 Copyright Holt McDougal. All rights reserved. esson 11.6 Geometry Notetaking Guide 319

8 Checkpoint Complete the following exercise. 3. Find the perimeter and area of the equilateral triangle inscribed in (F. Perimeter: about 31. units; Area: about 46.8 square units H F 6 6 G FINDING ENGTHS IN A EGUA N-GON To find the area of a regular n-gon with radius r, you may need to first find the apothem a or the side length s. You can use when you know n and as in... Pythagorean Theorem: Two measures: r and a, or r and s Example and Checkpoint Ex. 1 1 } s 1 a r Homework Special ight Triangles Any one measure: r or a or s And the value of n is 3, 4, or 6 Checkpoint Ex. 3 Trigonometry Any one measure: r or a or s Example 3 and Checkpoint Ex. 3 Copyright Holt McDougal. All rights reserved. esson 11.6 Geometry Notetaking Guide 319