The focus of this lesson is on applying the formula for finding the area of a regular polygon.
Center of a Regular Polygon
the center of the circumscribed circle
Radius of a Regular Polygon
the distance from the center to a vertex
Central Angle of a Regular Polygon
an angle formed by two radii drawn to consecutive vertices
Apothem of a Regular Polygon
the (perpendicular) distance from the center of a regular polygon to a side
The area of a regular polygon is equal to half the product of the apothem and the perimeter.
Area of a regular polygon
The area of a regular polygon is:
A = ½ Pa
The center of circle A is:
The center of pentagon BCDEF is:
A radius of circle A is:
A radius of pentagon BCDEF is:
An apothem of pentagon BCDEF is:
B C D E F A G
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
Areas of Regular Polygons Center of a regular polygon: center of the circumscribed circle. Radius: distance from the center to a vertex. Apothem: Perpendicular distance from the center to a side. Example 1: Find the measure of each numbered angle. • 1 2 3 360/5 = 72 ½ (72) = 36 L 2 = 36 L 3 = 54 L 1 = 72 Example 2: Find the area of a regular decagon with a 12.3 in apothem and 8 in sides. Area of a regular polygon: A = ½ a p where a is the apothem and p is the perimeter. Perimeter: 80 in A = ½ • 12.3 • 80 A = 492 in 2 Example 3: Find the area. 10 mm • A = ½ a p p = 60 mm 5 mm LL = √3 • 5 = 8.66 a A = ½ • 8.66 • 60 A = 259.8 mm 2
But what if we are not given any angles.
Ex: A regular octagon has a radius of 4 in. Find its area.
First, we have to find the apothem length.
4sin67.5 = a
3.7 = a
Now, the side length.
4 a 135 o 67.5 o 3.7 x 4cos67.5 = x 1.53 = x A = ½ Pa = ½ (24.48)(3.7) = 45.288 in 2
Central of a polygon – an whose vertex is the center & whose sides contain 2 consecutive vertices of the polygon.