Areas of Regular Polygons and Circles

– Find areas of regular polygons.– Find areas of circles.

AREAS OF REGULAR POLYGONSFirst, some definitions:

Regular Polygon – a polygon in which all segments and all angles are congruent.

Center of a Polygon – the center of its circumscribed circle

Radius of a polygon – the radius of its circumscribed circle, or the distance from the center to a vertex.

Apothem of a polygon – distance from the center to any side of the polygon.

AREAS OF REGULAR POLYGONS

Example:

Regular hexagon ABCDEF

A

B C

D

EF

Center and radius

Apothem

AREAS OF REGULAR POLYGONS

Example: regular hexagon

A

B C

D

EF

GNotice that triangle GFA is isosceles since all of the radii are congruent.

The area of the hexagon can be determined by adding the areas of the triangles.

AREAS OF REGULAR POLYGONS

Example: regular hexagon

A

B C

D

EF

Ga

b

Since the apothem is perpendicular to the side of the hexagon, it is an altitude to ∆AGF

Area of ∆AGF = ½ ba

Area of the hexagon is 6(½ ba)

AREAS OF REGULAR POLYGONS

Example: regular hexagon

A

B C

D

EF

Ga

b

Notice that the perimeter P of the hexagon is 6b units.

Area of the hexagon is 6(½ ba)

We can substitute P for 6b in the area formula.

Area of the hexagon is ½ Pa

Key Concept Area of a Regular Polygon

If a regular polygon has an area of A square units, a perimeter of P units, and an apothem of a units, then

A = ½Pa

This formula can be used to find the area of any regular polygon.

Example 1 Area of a Regular Polygon

Find the area of a regular pentagon with a perimeter of 40 centimeters. J

K

L

MN

P

Q

Step 1:

The internal angles of the pentagon add up to 360°, so …

Example 1 Area of a Regular Polygon

Find the area of a regular pentagon with a perimeter of 40 centimeters. J

K

L

MN

P

Q

Step 1:

The measure of each angle

Is or 72°360°5

PQ is the apothem of pentagon JKLMN. It bisects NPM and is a perpendicular bisector to NM. So MPQ is ½(72°) or 36°.

36°

Example 1 Area of a Regular Polygon

Find the area of a regular pentagon with a perimeter of 40 centimeters. J

K

L

MN

P

Q

Step 2:36°

Since the perimeter is 40 centimeters, each side is 8 centimeters and QM is 4 centimeters.

8

4

Example 1 Area of a Regular Polygon

J

K

L

MN

P

Q

36°8

4

Write a trigonometric ratio to find the length of PQ

5.536tan4

436tan)(

436tan

tan

PQ

PQ

PQ

PQ

PQQM

MPQ

Example 1 Area of a Regular Polygon

J

K

L

MN

P

Q

8

4

Area:

110

)5.5)(40(2121

PaA

5.5

Key Concept Area of a Circle

If a circle has an area of A square units and a radius of r units, then

A = πr2

r

Example 2 Use Area of a Circle toSolve Real World Problems

48 34

A caterer has a 48-inch table that is 34 inches tall. She wants a tablecloth that will touch the floor. Find the area of the tablecloth.

3.568,10

)58( 2

2

rA

Example 3 Area of an Inscribed polygon

Find the area of the shaded region. Assume the triangle is equilateral.

4