Areas of Regular Polygons Section 11.2

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. where ‘A’ represents the area and ‘s’ represents the side length of the triangle.

Example 1: Find area of the triangle. 10

Example 2: If the area of an equilateral triangle is 26cm 2, find the length of each side.

Other Regular Polygons: Center of Polygon: where diagonals intersect. Central angle of a polygon: an angle whose vertex is the center of the polygon and whose sides contain two consecutive vertices of the polygon:

Radius: the distance from center to vertex (corner) Apothem: the distance from the center of a polygon to any side.

Theorem 11.4: Area of a Regular Polygon: The area of a REGULAR n-gon, with side length s is half the product of the apothem a and the perimeter P, so where ‘A’ represents the area ‘a’ represents the apothem of the n-gon and ‘P’ represents the perimeter of the n-gon.

Example 3: A regular pentagon with radius 4. Find the area of the pentagon. a) What type of shape is this? _____________________ b) What is the central angle? _____________________ What is the top angle in the triangle? ________________ c) What is the apothem? _____________________ d) What is half the side length of the polygon? ___________ e) What is the whole side length of the polygon? __________ f) What is the perimeter of the polygon? _______________ g) Find the Area. __________________

Example 4: A regular hexagon is inscribed in a circle with apothem 5. Find the area of the hexagon. a) What type of shape is this? _____________________ b) What is the central angle? _____________________ What is the top angle in the triangle? ________________ c) What is the apothem? _____________________ d) What is half the side length of the polygon? ___________ e) What is the whole side length of the polygon? __________ f) What is the perimeter of the polygon? _______________ g) Find the Area. __________________