Regular Decagon — Definition, Formula & Examples

A regular decagon is a 10-sided polygon where every side has the same length and every interior angle has the same measure. Each interior angle in a regular decagon is 144°.

A regular decagon is an equilateral and equiangular polygon with 10 vertices, 10 sides of equal length, and 10 interior angles each measuring exactly $\frac{(10-2) \cdot 180°}{10} = 144°$ .

Key Formula

A = \frac{5s^2}{2}\tan(72°) \approx 7.694\,s^2

Where:

$A$ = Area of the regular decagon
$s$ = Length of one side

How It Works

To work with a regular decagon, you typically need to know its side length. From the side length, you can find the apothem (the distance from the center to the midpoint of a side), which lets you calculate the area. The sum of all interior angles is

1440°

, and since the shape is regular, each angle is

144°

. A regular decagon also has 35 diagonals, found using the diagonal formula

\frac{n(n-3)}{2}

with

n = 10

Worked Example

Problem: Find the area of a regular decagon with a side length of 6 cm.

Write the formula: Use the area formula for a regular decagon.

A = \frac{5s^2}{2}\tan(72°)

Substitute the side length: Plug in

s = 6

cm.

A = \frac{5(6)^2}{2}\tan(72°) = \frac{5 \cdot 36}{2} \cdot 3.0777 = 90 \cdot 3.0777

Calculate: Multiply to get the final area.

A \approx 276.99 \text{ cm}^2

Answer: The area of the regular decagon is approximately 277.0 cm².

Why It Matters

Regular decagons appear in architecture, tiling patterns, and coin designs — some coins around the world are decagonal. Understanding them reinforces the general polygon formulas for angles, area, and diagonals that you use throughout geometry.

Common Mistakes

Mistake: Confusing the interior angle (144°) with the exterior angle (36°).

Correction: The exterior angle of a regular decagon is

360° \div 10 = 36°

. The interior angle is

180° - 36° = 144°

. These two always add to 180°.