How to Calculate the Area of Regular Polygons

The area of a regular polygon is half the apothem times the perimeter. The apothem is the distance from the center to the middle of a side, and that formula works because the polygon splits into identical triangles.

• Regular Triangle (Equilateral): Area \(A = \frac{\sqrt{3}}{4} \times \text{side}^2\)

• Regular Quadrilateral (Square): Area \(A = \text{side}^2\)

• Regular Pentagon: Area \(A = \frac{5}{4} \times \frac{\text{side}^2}{\tan(\frac{180}{5})}\)

• Regular Hexagon: Area \(A = \frac{3\sqrt{3}}{2} \times \text{side}^2\)

Worked examples: area of a regular polygon

Practice questions on the area of regular polygons

1. Find the area of an equilateral triangle with a side length of \(9 \text{ cm}\).
2. What is the area of a square with a side of \(10 \text{ cm}\)?
3. Determine the area of a regular hexagon with each side measuring \(5 \text{ cm}\).
4. For a regular triangle (equilateral) with a side length of \(9 \text{ cm}\) and an apothem of \(7.79 \text{ cm}\), calculate its area.
5. What is the area of a square with a side of \(10 \text{ cm}\) and an apothem of \(7.07 \text{ cm}\)?
6. Determine the area of a regular pentagon with each side measuring \(11 \text{ cm}\) and an apothem of \(9 \text{ cm}\).

1. \( 35.07 \text{ cm}^2 \)
2. \( 100 \text{ cm}^2 \)
3. \( 64.95 \text{ cm}^2 \)
4. \( 105.21 \text{ cm}^2 \)
5. \( 141.4 \text{ cm}^2 \)
6. \( 247.5 \text{ cm}^2 \)

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