1. Areas of Regular Polygons and Circles
Find areas of regular polygons.
Find areas of circles.

2. AREAS OF REGULAR POLYGONS
First, 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.

3. AREAS OF REGULAR POLYGONS
Example:
Regular hexagon ABCDEF B C
Center and radius A D
Apothem F E

4. AREAS OF REGULAR POLYGONS
Example: regular hexagon B C
Notice that triangle GFA is isosceles since all of the radii are congruent. G A D
The area of the hexagon can be determined by adding the areas of the triangles. F E

5. AREAS OF REGULAR POLYGONS
Example: regular hexagon B C
Since the apothem is perpendicular to the side of the hexagon, it is an altitude to ∆AGF G A D a b E F
Area of ∆AGF = ½ ba
Area of the hexagon is 6(½ ba)

6. AREAS OF REGULAR POLYGONS
Example: regular hexagon B C
Notice that the perimeter P of the hexagon is 6b units. G A D a b
We can substitute P for 6b in the area formula. E F
Area of the hexagon is 6(½ ba)
Area of the hexagon is ½ Pa

7. 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.

8. Example 1 Area of a Regular Polygon
Find the area of a regular pentagon with a perimeter of 40 centimeters. K J L P
Step 1:
The internal angles of the pentagon add up to 360°, so … N Q M

9. Example 1 Area of a Regular Polygon
Find the area of a regular pentagon with a perimeter of 40 centimeters. K J L P
Step 1:
36°
The measure of each angle
Is or 72°
360° 5 N Q M
PQ is the apothem of pentagon JKLMN. It bisects NPM and is a perpendicular bisector to NM. So MPQ is ½(72°) or 36°.

10. Example 1 Area of a Regular Polygon
Find the area of a regular pentagon with a perimeter of 40 centimeters. K J L P
Step 2: 8
36°
Since the perimeter is 40 centimeters, each side is 8 centimeters and QM is 4 centimeters. 4 N Q M

11. Example 1 Area of a Regular PolygonK
Write a trigonometric ratio to find the length of PQ J L P 8
36° 4 N Q M

12. Example 1 Area of a Regular PolygonK
Area: J L P 8
5.5 4 N Q M

13. 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

14. Example 2 Use Area of a Circle to Solve Real World Problems
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. 48 34

15. Example 3 Area of an Inscribed polygon
Find the area of the shaded region. Assume the triangle is equilateral. 4