Areas of Regular Polygons Chapter 11, Section 2 June 22, 2016

Goals To find the area of an equilateral triangle. To find the area of a regular polygon, such as the area of a dodecagon in example 4.

Vocabulary

center of the polygonradius of the polygon apothem of the polygon. The center of the polygon and radius of the polygon are the center and radius of its circumscribed circle. The distance from the center to any side of the polygon is called the apothem of the polygon. Hexagon ABCDEFG has center G, radius GA, and apothem GH G center E FA B CD a H radius apothem

central angle of a regular polygon A central angle of a regular polygon is an angle whose vertex is the center and whose sides contain two consecutive vertices of the polygon. Hexagon ABCDEFG has central angle AGB G center E FA B CD a H

Area of an Equilateral Triangle Theorem The area of an equilateral triangle is one fourth the square of the length of the side times. A = s 2

Example 1 - Proof of Theorem Prove that the area of an isosceles triangle with base b, and base angles that measure 45º is one-fourth the square of the base. PS R Q 45º b

Example 2 – Finding the Area of an Equilateral Triangle Find the area of an equilateral triangle with sides measuring 10 cm.

Area of a Regular Polygon Theorem s a P ns The area of a regular n-gon with side length s is half the product of the apothem a and the perimeter P (the product of the number of sides n and the side length s). A = aP or A = ans

How to Find the Area 1 st -Find the Central Angle Measure 360/n 2 nd -Draw the apothem (if needed) and find the measure of a & s. Special Rt. Triangles or SOH CAH TOA 3 rd -Find the Perimeter P=ns 4 th -Plug into the Area Formula A=1/2aP

Example 3 – Finding the Area of a Regular Polygon A regular Hexagon is inscribed in a circle with radius 4 units. Find the area of the hexagon.

Example 4 – Finding the Area of a Regular Dodecagon The bottom of a glass is a regular 12-gon with a side length of about 1.2cm and a radius of 2.3cm. What is the area of the bottom of the glass?

Homework Sec 11.2a Pg 672 #’s 1-7, 9-15 Sec 11.2b Pg 672 #’s 16-24, 30-32, 54-64

Homework P #1-7, 9-15 P 186 #2-34 every other even