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11.2 Areas of Regular Polygons

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11.2 Areas of Regular Polygons 10.3 Areas of Regular Polygons Geometry Mr. Peebles Spring 2013 Daily Learning Target (DLT) Monday January 14, 2013 • “I can remember, apply, and understand to find the perimeter and area of a regular polygon.” 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 s2 s A = ¼ s2 Ex. 1: Finding the area of an Equilateral Triangle • Find the area of an equilateral triangle with 8 inch sides. 2 A = ¼ s Area3 of an equilateral Triangle A = ¼ 82 Substitute values. A = ¼ • 64 Simplify. A = • 16 Multiply ¼ times 64. A = 16 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. 2 A = ¼ s Area3 of an equilateral Triangle A = ¼ 122 Substitute values. A = ¼ • 144 Simplify. A = • 36 Multiply ¼ times 144. A = 36 Simplify. Using a calculator, the area is about 62.4 square inches. More . • The apothem is the F A height of a triangle between the center H and two consecutive a E vertices of the G B polygon. • You can find the area of any regular n-gon by dividing the D C polygon into Hexagon ABCDEF with center G, radius GA, congruent triangles. and apothem GH More . A = Area of 1 triangle • # of triangles OR F A = ½ • apothem • # of sides • side length OR H a E = ½ • apothem • perimeter of a polygon 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 The number of congruent triangles formed will be the same as the number of A = ½ aP, or A = ½ a • ns. 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 S P = 12(4.3) = 51.6 feet 8.3 ft. A B Solution: S • In ∆SBT, BT = ½ (BA) = ½ (4.3) = 2.15 8.3 feet feet. Use the Pythagorean 2.15 ft. Theorem to find the A B apothem ST. T 4.3 feet a = 8.32 2.152 a 8 feet So, the floor area of the enclosure is: A = ½ aP ½ (8)(51.6) = 206.4 ft. 2 Assignment: 10-3 Pages 548-551 (4-9, 11, 24) Due Tomorrow Tuesday January 15, 2013 Closure On the whiteboards, write down in your own words what an Apothem is. .
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