A worked example: a regular decagon with 2-unit sides
Enter 10 sides and a side length of 2 into the Polygon Calculator and it returns a perimeter of 20, an area of 30.776835 square units, an interior angle of 144 degrees, and a sum of interior angles of 1440 degrees. The perimeter is the easy part — 10 sides of length 2 simply add up to 20 — but the area and angle both depend on the shape closing up into exactly ten equal corners.
The formula: perimeter, area, and interior angle for n sides
The calculator uses perimeter = n × s, area = (n × s²) / (4 × tan(π/n)), and interior angle = (n − 2) × 180 / n. Substituting n = 10 and s = 2 gives perimeter = 10 × 2 = 20, area = (10 × 2²) / (4 × tan(π/10)) = 30.776835, and interior angle = (10 − 2) × 180 / 10 = 144. The (n − 2) term counts how many triangles the polygon splits into from one corner, which is why it drives both the interior angle and the 1440-degree sum.
A smaller regular pentagon for comparison
Swap in 5 sides at a side length of 4 and the calculator returns a perimeter of 20 again — the same total perimeter as the decagon above, just distributed across fewer, longer sides — but a smaller area of 27.527638 square units and a smaller interior angle of 108 degrees, with a sum of interior angles of 540 degrees. Matching perimeters do not mean matching areas: spreading the same total length across more, shorter sides packs in more area, which is exactly what happens between the pentagon and the decagon.
Why interior angle formulas matter for design and construction
Anyone laying out a multi-sided deck, tiling a floor, or cutting frame pieces for a polygonal window needs the interior angle before cutting a single piece, since it sets the miter angle at every joint. Getting it from the formula rather than a protractor means the angle is exact, and it scales instantly to any number of sides without re-measuring anything by hand.