Calculate area, perimeter, apothem, circumradius, inradius, interior angle, exterior angle, and diagonals for any regular polygon from 3 to 20 sides. Comparison table of all shapes included.
A regular polygon is a polygon where all sides have equal length and all interior angles are equal. Examples range from the familiar equilateral triangle (3 sides) through the square (4), pentagon (5), hexagon (6), and on to dodecagons (12), icosagons (20), and beyond. As the number of sides increases, a regular polygon approaches a circle.
The key measurements of a regular n-gon with side length s are all derivable from n and s. The apothem (distance from center to midpoint of a side) is a = s / (2 tan(π/n)). The circumradius (center to vertex) is R = s / (2 sin(π/n)). The area is A = (n × s × a) / 2 = (n × s²) / (4 tan(π/n)). The perimeter is simply P = n × s.
Interior angles of a regular n-gon are each (n−2) × 180° / n, and exterior angles are 360° / n. The number of diagonals is n(n−3)/2 — a pentagon has 5, a hexagon has 9, a decagon has 35.
Regular polygons are fundamental in tiling and tessellation (only triangles, squares, and hexagons tile the plane), architecture (the Pentagon building), engineering (hex bolts, stop signs), nature (honeycombs, snowflake symmetry), and game design (dice shapes, hex grids).
This calculator works for any regular polygon from 3 to 20 sides. Enter the number of sides and the side length, and it computes every geometric property. A comparison table shows how these properties change as n increases, illustrating the convergence toward a circle.
This calculator is useful when you want to explore how one side length and one side count determine an entire family of regular-polygon measurements. Instead of working through trigonometric formulas separately for apothem, circumradius, area, and angles, you can change n or the side length once and see every property update together. That makes it practical for classroom geometry, layout work, game-board design, signage, tiling studies, and any project where symmetry and equal sides matter.
Interior Angle: (n−2) × 180° / n
Exterior Angle: 360° / n
Apothem: a = s / (2·tan(π/n))
Circumradius: R = s / (2·sin(π/n))
Perimeter: P = n × s
Area: A = (n × s²) / (4·tan(π/n)) = ½ × P × a
Diagonals: n(n−3)/2Result: Area ≈ 259.81 cm², Perimeter = 60 cm, Apothem ≈ 8.66 cm, Circumradius = 10 cm
For a regular hexagon with side length 10 cm, the perimeter is 6 × 10 = 60 cm. The apothem is 10 ÷ (2 tan(30°)) ≈ 8.66 cm, and the circumradius is 10 ÷ (2 sin(30°)) = 10 cm. Using A = 1/2 × perimeter × apothem gives an area of about 259.81 cm², while each interior angle is 120°.
A regular polygon is a good example of how symmetry reduces complexity. Once the side count and side length are known, every other major property follows from the same center-based geometry. The apothem, circumradius, central angle, interior angle, and area are all connected through congruent isosceles triangles formed by joining the center to adjacent vertices. This calculator exposes those relationships directly, which makes it easier to see the structure behind the formulas instead of treating each result as unrelated.
As the number of sides increases, a regular polygon more closely approximates its circumscribed circle. The comparison table and circle-fill output make that idea concrete. A triangle leaves a lot of unused circle area, a hexagon already fits much more tightly, and a 20-gon is visually close to circular. That progression matters in numerical methods, design approximations, and geometry lessons that connect polygons to limits, trigonometry, and the geometry of circles.
Regular polygons appear in floor tiling, fastener heads, road signs, decorative patterns, board-game grids, and architectural motifs. Designers care about perimeter and area, fabricators may care about across-corner size through the circumradius, and students often need angle and diagonal counts for proofs or problem sets. Because the calculator handles every regular n-gon from 3 to 20 sides, it also works as a fast comparison tool when deciding which polygon best fits a visual or structural requirement.
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Interior angle = (n−2) × 180° / n, where n is the number of sides. A hexagon has 120°, an octagon has 135°.
The apothem is the perpendicular distance from the center to the midpoint of any side. It equals s / (2·tan(π/n)).
The formula is n(n−3)/2. A pentagon has 5, a hexagon has 9, a decagon has 35.
The circumradius R is the distance from center to a vertex (outer circle). The inradius (apothem) is the distance from center to the midpoint of a side (inner circle).
Only equilateral triangles, squares, and regular hexagons can tile the plane edge-to-edge by themselves. Their interior angles divide evenly into 360°.
For a fixed side length, area increases with n because the polygon fills more of its circumscribed circle. A 20-gon has about 98.8% of its circumscribed circle's area.
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