10.3 Areas of Regular Polygons Geometry Mr. Peebles Spring 2013 Bell Ringer • Find the area of an equilateral triangle with 6 inch sides. PLEASE LEAVE YOUR ANSWER IN RADICAL FORM. A = ¼ 3 s2 Area of an equilateral Triangle Bell Ringer • Find the area of an equilateral triangle with 6 inch sides. PLEASE LEAVE YOUR ANSWER IN RADICAL FORM. A = ¼ 3 s2 Area of an equilateral Triangle A = ¼ 3 62 Substitute values. A = ¼ 3 • 36 Simplify. A= Multiply ¼ times 36. 3 •9 A=9 3 Simplify. Daily Learning Target (DLT) Tuesday January 15, 2013 • “I can remember, apply, and understand to find the perimeter and area of a regular polygon.” Assignment: 10-3 Pages 548-551 (4-9, 11, 24) 4. 5. 6. 7. 8. 2144.475 cm2 2851.8 ft2 12,080 in2 2475 in2 1168.5 m2 9. 2192.4 cm2 11. 27.7 in2 24. D Finding the area of an equilateral triangle • The area of any triangle with base length b and height h is given by A = ½bh. The following formula for equilateral triangles; however, uses ONLY the side length. * Look at the next slide * Theorem 11.3 Area of an equilateral triangle • The area of an equilateral triangle is one fourth the square of the length of the side times 3 s s A = ¼ 3 s2 s A = ¼ 3 s2 Ex. 1: Finding the area of an Equilateral Triangle • Find the area of an equilateral triangle with 8 inch sides. A = ¼ 3 s2 Area of an equilateral Triangle A = ¼ 3 82 Substitute values. A = ¼ 3 • 64 Simplify. A= Multiply ¼ times 64. 3 • 16 A = 16 3 Simplify. Using a calculator, the area is about 27.7 square inches. Ex. 2: Finding the area of an Equilateral Triangle • Find the area of an equilateral triangle with 12 inch sides. A = ¼ 3 s2 Area of an equilateral Triangle A = ¼ 3 122 Substitute values. A = ¼ 3 • 144 Simplify. A= Multiply ¼ times 144. 3 • 36 A = 36 3 Simplify. Using a calculator, the area is about 62.4 square inches. More . . . • The apothem is the height of a triangle between the center and two consecutive vertices of the polygon. • You can find the area of any regular n-gon by dividing the polygon into congruent triangles. F A H a E G D B C Hexagon ABCDEF with center G, radius GA, and apothem GH More . . . F A = Area of 1 triangle • # of triangles OR A = ½ • apothem • # of sides • side length OR = ½ • apothem • perimeter of a polygon H a E G B This approach can be used to find the area of any regular polygon. D C Hexagon ABCDEF with center G, radius GA, and apothem GH Theorem 11.4 Area of a Regular Polygon • The area of a regular n-gon with side lengths (s) is half the product of the apothem (a) and the perimeter (P), so A = ½ aP, or A = ½ a • ns. The number of congruent triangles formed will be the same as the number of sides of the polygon. NOTE: In a regular polygon, the length of each side is the same. If this length is (s), and there are (n) sides, then the perimeter P of the polygon is n • s, or P = ns Example 3 Find the area of a regular pentagon with an apothem of 4 Feet and side length of 3 Feet. = ½ • apothem • # of sides • side length OR = ½ • apothem • perimeter of a polygon Apothem = 4 Feet Numbers of Sides = 5 Side Length = 3 Example 3 Find the area of a regular pentagon with an apothem of 4 Feet and side length of 3 Feet. = ½ • apothem • # of sides • side length OR = ½ • apothem • perimeter of a polygon Apothem = 4 Feet Numbers of Sides = 5 Side Length = 3 So ½•4•3•5 = 30 Square Feet Example 4 Find the area of a regular octagon STOP SIGN with an apothem of 9 Feet and side length of 12 Feet. = ½ • apothem • # of sides • side length OR = ½ • apothem • perimeter of a polygon Apothem = 9 Feet Numbers of Sides = 8 Side Length = 12 Example 4 Find the area of a regular octagon STOP SIGN with an apothem of 9 Feet and side length of 12 Feet. = ½ • apothem • # of sides • side length OR = ½ • apothem • perimeter of a polygon Apothem = 9 Feet Numbers of Sides = 8 Side Length = 12 So ½ • 9 • 8 • 12 = 432 Square Feet Ex. 5: Finding the area of a regular dodecagon • Pendulums. The enclosure on the floor underneath the Foucault Pendulum at the Houston Museum of Natural Sciences in Houston, Texas, is a regular dodecagon with side length of about 4.3 feet and a radius of about 8.3 feet. What is the floor area of the enclosure? Solution: • A dodecagon has 12 sides. So, the perimeter of the enclosure is P = 12(4.3) = 51.6 feet S 8.3 ft. A B Solution: S • In ∆SBT, BT = ½ (BA) = ½ (4.3) = 2.15 feet. Use the Pythagorean Theorem to find the apothem ST. a= 8.3 2.15 2 2 8.3 feet 2.15 ft. A T 4.3 feet a 8 feet So, the floor area of the enclosure is: A = ½ aP ½ (8)(51.6) = 206.4 ft. 2 B Assignment: 10-3-Test Prep Pages 548-551 Due Today Tuesday January 15, 2013 39, 40, 44, 45, 46 Exit Quiz – 5 Points • Find the area of an equilateral triangle with 10 inch sides. PLEASE LEAVE YOUR ANSWER IN RADICAL FORM. A = ¼ 3 s2 Area of an equilateral Triangle
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