Areas of Regular Polygons

Section 11.2 - Areas of Regular Polygons

Theorem 11.3 – Area of an Equilateral Triangle

·  The area of an equilateral triangle is one fourth the square of the length of the side times.

A = ¼s2

Finding the Area of a Regular Polygon

Think of the hexagon below as inscribed in a circle.

The center of the polygon and radius of the polygon are the center and radius of the circumscribed circle.

Apothem of a polygon – The (perpendicular) distance from the center to any side of the polygon, and is also the height of a triangle between the center and two consecutive vertices of the polygon

The area of an n-gon is found by dividing the polygon into congruent triangles.

A = (area of one triangle)(number of triangles)

A = (1/2)(apothem)(side length s)(number of sides)

A = (1/2)(apothem)(number of sides)(side length s)

A = 1/2(apothem)(perimeter of polygon)

Theorem 11.4 – Area of a Regular Polygon

·  The area of a regular n-gon with side length s is half the product of the apothem (a) and the perimeter (P), so…

A = ½aP or A ½a(ns)

Central Angle of a Regular Polygon – is a angle whose vertex is the center and whose sides contain two consecutive vertices of the polygon.

To find the area of a regular polygon, follow these steps:

1.  Remember the area formula: A = ½aP

2.  Determine what information you need to find (not given in the problem) to plug into the formula

·  apothem (a)

·  Side length of the polygon (so that you can find the Perimeter)

3.  Pull a right triangle out of the polygon bounded by a radius and an apothem and sketch.

Ø Find the central angle measure of the polygon (360/n) and divide it in half – this is one angle in your pulled out triangle

Ø Use the trig ratios (SOH CAH TOA) or special right triangles (30-60-90 or 45-45-90) to find the apothem or side length of the small triangle that you pulled out

Ø Remember to double the side length of your pulled out triangle to get the full side length of the whole polygon

4.  Calculate the Perimeter by taking the side length of the polygon and multiplying it by the number of sides

5.  Plug the apothem and Perimeter into the area formula.