Elena is planning a triangular garden. She wants to build a fence around the garden to keep out the deer. The length of one side of the garden is 26 feet. If the angles at the end of this side are 780 and 440, find the length of fence needed to enclose the garden.

In quadrilateral ABCD, AC = 188, BD = 214, m∠BPC = 70, and P is the midpoint of segment AC and segment BD. Find the perimeter of quadrilateral ABCD.

Warm-up

A B

CD

P

Angle Measures in Polygons

The sum of the measures of the interior angles of a convex n-gon (n-2)∙1800.

Corollary to Theorem 11.1The measure of each interior angle of a

regular n-gon is

Theorem 11.1 - Polygon Interior Angles Theorem

€

1

n• (n − 2) • 180o or

(n - 2) • 180o

n

Theorem 11.2 – The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is 3600.

Corollary to Theorem 11.2

The measure of each exterior angle of a regular n-gon is

Theorems about the exterior angles

€

1

n• 360o, or

360o

n

1

2

5

43

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 time .

Area of Regular Polygons

€

3

€

A =1

43s2

ss

s

How many triangles are formed? What kind of triangles are they?

Find the area of one of the triangles. What is the area of the entire hexagon? Explain.

Investigate Regular Hexagon

Theorem 11.4The area of a regular n-gon with side length s is half the product of the apothem a and the perimeter P, so A = ½ aP, or A = ½ a ∙ ns

Area of a Regular Polygon