Area of Regular Polygons — Practice Geometry Questions

If you are asked to find the area of a regular polygon, you can do so by using a formula that includes the perimeter of the polygon and a measurement called the apothem.

The apothem is the line segment from the center of the polygon to the midpoint of one of the sides, and is perpendicular to that side. The perimeter is the total distance around the polygon.

The formula for the area of a regular polygon is

Practice questions

1. Find the area of a regular pentagon whose perimeter is 40 units and whose apothem is 5 units.

2. Find the exact area of a regular hexagon that has a perimeter of 60 units.

Answers and explanations

1. 100 units²

   The formula for the area of a regular polygon is

   

   The apothem is 5 and the perimeter is 40, so the area is

   

2. 

   The formula for the area of a regular polygon is

   

   A regular hexagon is a polygon with six equal sides. You're given that the perimeter of the hexagon is 60 units, which means each side is 10. The apothem is joined to the midpoint of one of the sides and is also perpendicular to the side, forming a

   

   The side opposite the 30-degree angle is x, the side opposite the 60-degree angle is

   

   and the side opposite the 90-degree angle is 2x. The apothem is opposite the 60-degree angle, so the apothem equals

   

   When you plug everything into the formula, you get