Area of a Regular Polygon
Area from apothem and perimeter.
Formula
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Example
Hexagon, apothem 5, perimeter 34.6: A = ½·5·34.6 ≈ 86.6.
How it works
A regular polygon can be split into equal isosceles triangles by drawing lines from its center to each vertex. The apothem is the perpendicular distance from the center to a side, so each triangle has height a and all of their bases add up to the full perimeter P. Since the total area is the sum of the triangle areas, the formula becomes A = 1/2 × a × P.
When to use it
Use this formula when you know the apothem and total perimeter of a regular polygon such as a regular pentagon, hexagon, or octagon. It is especially useful in geometry problems, design work, and any situation where a shape has equal sides and equal angles.
Worked example, step by step
A regular octagon has apothem 7 cm and perimeter 40 cm. A = 1/2 × a × P A = 1/2 × 7 × 40 A = 3.5 × 40 A = 140 So, the area is 140 cm².
Common mistakes to avoid
- Using the side length instead of the apothem
- Forgetting to use the full perimeter of the polygon
- Leaving out the 1/2 factor in the formula
Did You Know?
The apothem — the distance from center to a side's midpoint — is a polygon's version of a radius.
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