What Is a Decagon?
A decagon is a polygon with ten sides and ten angles. A regular decagon has all sides equal in length and all interior angles equal (each interior angle measures 144°). This calculator computes the area of a regular decagon directly from a single measurement: the length of one side.
How to Use This Calculator
Enter the length of one side (s) of your regular decagon in any unit you like — centimeters, inches, meters. The result is returned in the matching square units. The tool also reports the perimeter, which is simply ten times the side length.
The Formula Explained
The area of a regular decagon is given by:
$$A = \frac{5}{2} \cdot \frac{\text{Side (s)}^{2}}{\tan\left(\frac{\pi}{10}\right)}$$
Here \(\pi/10\) radians equals 18°. The factor \(\tan(18°) \approx 0.32492\). Because the decagon can be divided into ten congruent isosceles triangles meeting at the center, the apothem-based area formula simplifies to this neat closed form. The numerical coefficient \((5/2)/\tan(18°) \approx 7.694\), so the area is approximately \(7.694 \cdot s^{2}\).
Worked Example
Suppose a regular decagon has a side length of 10 units. Then:
$$A = 7.694 \times 10^{2} = 7.694 \times 100 \approx 769.42 \text{ square units}$$ Its perimeter is \(10 \times 10 = 100\) units.
FAQ
What is the area coefficient for a decagon? Approximately 7.69420884. Multiply this by the square of the side length to get the area.
Does this work for irregular decagons? No. This formula assumes a regular decagon with equal sides and angles. For irregular shapes, split the figure into triangles.
What units does it use? Whatever unit you enter for the side; the area comes out in those units squared.
