Find the area, perimeter, and apothem of any regular polygon, from triangles to dodecagons, with a live-drawn diagram.
Choose the type of regular polygon (a shape where every side and every angle is equal) from the dropdown, then enter the length of one side. The diagram above the form redraws itself automatically so you can see exactly which shape you're working with. Press "Calculate Area & Perimeter" to get the area, total perimeter, apothem (the distance from the center to the middle of a side), and interior angle.
For a regular polygon with n sides of length s:
Perimeter = n × s
Apothem = s ÷ (2 × tan(π / n))
Area = (n × s × apothem) ÷ 2, which simplifies to (n × s²) ÷ (4 × tan(π/n))
Interior Angle = ((n − 2) × 180°) ÷ n
For a regular hexagon (n = 6) with a side length of 4 cm: perimeter = 6 × 4 = 24 cm. The apothem = 4 ÷ (2 × tan(30°)) ≈ 3.46 cm. Area = (6 × 4 × 3.46) ÷ 2 ≈ 41.57 cm². Each interior angle = ((6−2) × 180) ÷ 6 = 120°. This matches what the calculator returns when you select Hexagon and enter a side length of 4.
Regular polygon formulas show up in carpentry (cutting hexagonal tabletops), tiling and paving patterns, architecture (octagonal rooms or gazebos), and geometry homework. Because the shape redraws live as you change the number of sides, it also works well as a quick visual reference for what each polygon actually looks like.
Does this work for irregular polygons? No — this calculator assumes a regular polygon where all sides and angles are equal. Irregular shapes require different methods depending on their exact vertices.
What's the apothem used for? The apothem is a key measurement in construction and design since it defines the "radius" of the largest circle that fits perfectly inside the polygon.
Area = (1/4) × n × side² × cot(π/n), where n is the number of sides.
Perimeter = n × side length.
All sides and all interior angles are equal.