11.5 Areas of Regular Polygons

Equilateral Triangle

Remember: drop an altitude and you create two 30-60-90 triangles.

What is the measure of the sides and altitude in terms of one side equaling s?

Given: ∆ CAT is equilateral, and TA = s

Find the area of ∆CAT T A

C

S

Find the altitude (h) of the ∆.

A ∆CAT =

=

=

1

2s(s

23)

1

2bh

S

43

2

T106: Area of an equilateral triangle = the product of 1/4 the square of a side and the square root of 3. Where s is the length of a side

Aeq∆ =

S

43

2

Area of a regular polygon:

Remember all interior angles are congruent and all sides are equal.Regular pentagon:

•O is the center

•OA the radius

•OM is an apothem

N

T

A

O

M

E

P

You can make 5 isosceles triangles in a pentagon.

Any regular polygon:

Radius: is a segment joining the center to any vertex

Apothem: is a segment joining the center to the midpoint of any side.

See page 532 for observationsApothems:

1.Congruent only in regular polygons.

2.Radius of a circle inscribed in a polygon.

3.Perpendicular bisector of a side.

4.A radius of a regular polygon bisects an angle of the polygon.

5.If all radii are drawn, the polygon is divided into congruent isosceles triangles.

6.The altitude of each triangle is its apothem.

T107: Areg poly = 1/2 ap

Area of a regular polygon equals one-half the product of the apothem and the perimeter.

Where a = apothem

p = perimeter

Find the area of a regular hexagon whose sides are 18cm long.

1.Draw the picture

2.Write the formula

3.Plug in the numbers

4.Solve and label units

18cm

Find the perimeter

Find each angle

Find the apothem

Write the formula, and solve.