Formula and Method for the Decagon Calculator
A regular decagon is a 10-sided polygon with all sides equal in length and all interior angles equal to 144°. Because it is regular, every property can be derived from a single measurement — the side length s. This calculator splits the decagon into 10 congruent isosceles triangles that meet at the center, each with a central angle of 360°/10 = 36°, to compute the area, perimeter, apothem (inradius), and circumradius, plus an optional material cost estimate.
How the calculation works
Enter the side length and choose the unit it is measured in. Each of the 10 triangles has a base of s and a central angle of 36°, so its half-angle is 18°. The perpendicular height from the center to the base — the apothem — is a = s / (2tan(18°)), and the distance from the center to each vertex — the circumradius — is R = s / (2sin(18°)). The area of the whole decagon is 10 times the area of one triangle, which simplifies to A = (5/2)s²cot(18°) ≈ 7.6942 × s². As a check, the same area also equals half the perimeter times the apothem: A = (1/2) × (10s) × a. The perimeter is simply P = 10s. If you enter a cost per square unit, the tool multiplies it by the area to estimate total material cost.
Common mistakes
- Confusing side length with circumradius: the circumradius (s/(2sin18°) ≈ 1.618s) is always longer than the side itself — do not substitute one for the other.
- Mixing units: keep the side length in one consistent unit — convert inches to feet, or centimeters to meters, before entering the value.
- Assuming an irregular decagon: these formulas apply only to a regular decagon (equal sides and angles); an irregular 10-sided shape needs coordinate geometry or triangulation instead.
Real-world applications
- Woodworking, tiling, and fabrication projects use the area directly to estimate material needed for decagonal tabletops, tiles, or gazebo floors.
- Framing and layout use the perimeter to determine how much edge trim, molding, or fencing material is required.
- CNC and CAD layouts use the circumradius to size the bounding circle a decagon must fit inside, and the apothem to check clearance to each flat side.
- Architecture and coin/medallion design use decagons for their near-circular, ten-fold symmetric silhouette, which is why several circulating coins (such as the US Susan B. Anthony dollar's edge and some Canadian coins) use decagon or similar polygon-based shapes.