The area formula
A = perimeter × apothem ÷ 2
Where the formula comes from
Draw lines from the centre to every vertex and the polygon becomes n identical triangles, each with base equal to a side and height equal to the apothem. Each has area (side × apothem)/2, and n of them give (perimeter × apothem)/2. As n grows the apothem approaches the radius and the perimeter approaches the circumference, so the formula becomes πr² in the limit.
A worked example, step by step
A hexagon with sides of 4 cm
- Perimeter: 6 × 4 = 24 cm.
- Apothem: 4 ÷ (2 tan(30°)) = 3.46 cm.
- Area: 24 × 3.46 ÷ 2 = 41.57 cm².
- Interior angle: (6 − 2) × 180 ÷ 6 = 120°.
Watch the units: area is always in square units.
Where it turns up in practice
Tiling, nuts and bolts, paving and structural design. The interior angle decides whether a shape tiles the plane: only the triangle, the square and the hexagon have angles that divide 360° exactly, which is why you never see a regular pentagon floor.