How to Calculate the Area of Regular Polygons

Step-by-step Guide: Area of Regular Polygons

Using Side Length:

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

Using Apothem:

For a regular polygon with \( n \) sides, each of length \( s \), and an apothem of length \( a \):

Area \(A = \frac{n \times s \times a}{2}\)

Examples

Example 1:
Calculate the area of an equilateral triangle with a side length of \(6 \text{ cm}\).

Solution:
\( A = \frac{\sqrt{3}}{4} \times 6^2 \text{ cm}^2 = 15.59 \text{ cm}^2 \)

Example 2:
Find the area of a square with a side of \(5 \text{ cm}\).

Solution:
\( A = 5^2 \text{ cm}^2 = 25 \text{ cm}^2 \)

Example 3:
Determine the area of a regular pentagon with a side length of \(8 \text{ cm}\).

Solution:
\( A = \frac{5}{4} \times \frac{8^2 \text{ cm}^2}{\tan(36^\circ)} = 110.11 \text{ cm}^2 \)

Example 4:
Given a regular hexagon with a side of \(7 \text{ cm}\), compute its area.

Solution:
\( A = \frac{3\sqrt{3}}{2} \times 7^2 \text{ cm}^2 = 127.28 \text{ cm}^2 \)

Example 5:
Calculate the area of a regular pentagon given an apothem length of \(7 \text{ cm}\) and a side length of \(8 \text{ cm}\).

Solution:
\( A = \frac{5 \times 8 \text{ cm} \times 7 \text{ cm}}{2} = 140 \text{ cm}^2 \)

Example 6:
Determine the area of a regular hexagon with an apothem of \(6 \text{ cm}\) and each side measuring \(7 \text{ cm}\).

Solution:
\( A = \frac{6 \times 7 \text{ cm} \times 6 \text{ cm}}{2} = 126 \text{ cm}^2 \)

Practice Questions:

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 \(9 \text{ cm}\).

Answers:

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. \( 247.5 \text{ cm}^2 \)