Decagon Area Calculator

A regular decagon features ten equal sides and ten equal interior angles. The core formula used to find the area of a regular decagon based on its side length (a) is derived by splitting the polygon into ten identical isosceles triangles meeting at the center. The mathematical expression is: Area = (10 × a^2 × cot(π / 10)) / 4. Alternatively, you can determine the area using the perimeter (P) and the incircle radius (r, or apothem) via the general polygon formula: Area = (P × r) / 2. Other key metrics include the perimeter (P = 10 × a), the circumcircle radius, and the interior angles, which always sum up to 1440 degrees total, with each individual interior angle measuring 144 degrees.

Let us calculate the total area and perimeter of a regular decagon with a side length (a) of 3 cm. First, find the perimeter by multiplying the side length by the number of sides: Perimeter = 10 × 3 = 30 cm. Next, apply the decagon area formula: Area = (10 × 3^2 × cot(π / 10)) / 4. Since 3^2 equals 9, the numerator becomes 90 × cot(18°). Knowing that cot(18°) is approximately 3.07768, we multiply 90 by 3.07768 to get 276.9912. Finally, dividing this product by 4 yields an area of approximately 69.25 square centimeters.

When working with regular polygons like the decagon, always double-check whether your angle measurements are set to degrees or radians before applying trigonometric functions like sine, tangent, or cotangent. Keep in mind that the apothem (incircle radius) is always shorter than the circumradius because it extends from the center to the midpoint of a side, whereas the circumradius reaches the outer vertices. Utilizing these relationships correctly prevents compounding rounding errors in multi-step architectural or engineering designs.