Areas of Regular Polygons

10-3

Honor’s

On a sheet of warm up paper:• Write the name of your podcast

group members (don’t write your own name)

• Rate each member from 1-10, 10 being “very helpful”

Area Formulas

Rhombus: .5(D1)(D2)

Trapezoid: .5(h)(b1+b2)

Kite: There ain’t one, use your common sense!

Definitions

• Center – the center of the circle circumscribed about the polygon

• radius – a segment drawn from the center of a polygon to a vertex

• apothem – a segment drawn from the center of a polygon that is perpendicular to a side

• central angle – an angle formed by two radii drawn to consecutive vertices

radi

us

center

apothem

Central angle

Theorem 11.6 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

A = ½ aP, or A = ½ a • ns.

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

The number of congruent triangles formed will be the same as the number of sides of the polygon.

More . . .

• 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. You can divide 360° by the number of sides to find the measure of each central angle of the polygon.

• 360/n = central angle

Ex: Finding the area of a regular polygon

• A regular pentagon with radius 1 unit. Find the area of the pentagon.

B

C

A

1

1D

Solution:

• you must find the apothem (or if the apothem was given, you must find the radius, etc)

• You need to find measure of central angle. ABC is 360°/5, or 72°.

1

DA C

B

Solution:

• Draw the apothem. It is an isosceles triangle so it bisects the angle.

• You now have a right triangle and can use trig ratios to find the missing sides

1

DA C

B

36°

Solution

1

DA C

B

You try…..

• Find the area of a regular polygon with 9 sides and a radius of 10