Area of Regular Polygons Using Apothem

Area of a Regular Polygon Using the Apothem

A regular polygon has all sides and angles equal. When its apothem is given, the area can be found by using the perimeter and the apothem together.

Method

1. Find the perimeter of the polygon.
   • If one side length is given, multiply it by the number of sides.
2. Use the area idea for regular polygons:
   • Area = half of the apothem times the perimeter.
3. Substitute the values carefully and simplify.
   • If the answer involves radicals or fractions, reduce it fully.

Why this works

A regular polygon can be split into congruent triangles from the center. The apothem is the height of each triangle, so the total area is the sum of those triangle areas. That leads to the compact relationship between apothem and perimeter.

Check your work

• Make sure the perimeter matches the number of sides.
• Confirm the apothem and perimeter use the same units.
• The final area should always be in square units.

Quick strategy

If you are unsure, first compute the perimeter, then multiply by the apothem, and divide by 2. This gives a reliable result every time.