Regular Polygon Calculator

A regular polygon has all sides equal and all angles equal, like a square or a regular hexagon. This calculator takes the number of sides and the side length and finds the interior and exterior angles, the sum of the angles, the perimeter, the apothem, the circumradius, the area and the number of diagonals, and draws the shape.

Try:

How to use the regular polygon calculator

  1. Enter the number of sides (3 or more).
  2. Enter the side length.
  3. Press Calculate to see the angles, area and a drawing.

Formula

Interior angle=(n−2)×180∘n\text{Interior angle} = \frac{(n-2) \times 180^\circ}{n}
A=ns24tan⁡(π/n)A = \frac{n s^2}{4\tan(\pi/n)}

Worked example: Hexagon, side 4

  • Number of sides: 6
  • Side length: 4

✓ Answer checked

Interior angle=120∘,Area≈41.569219\text{Interior angle} = 120^\circ,\quad \text{Area} \approx 41.569219
Shape
Regular hexagon
Sum of interior angles
720°
Exterior angle
60∘60^\circ
Perimeter
2424
Apothem / circumradius
3.4641  /  43.4641 \;/\; 4
Diagonals
9
A regular hexagon with side 4. The dashed line is the apothem (3.4641).
Step-by-step working (3 steps)

  1. Interior angles Angle sum

    The angles of an n-sided polygon add to (n − 2) × 180°; in a regular polygon they are all equal.

    (6−2)×180∘6=120∘\frac{(6 - 2) \times 180^\circ}{6} = 120^\circ

  2. Apothem

    The distance from the centre to the middle of a side.

    a=s2tan⁡(180∘/n)≈3.4641016a = \frac{s}{2\tan(180^\circ/n)} \approx 3.4641016

  3. Area A = ½ × perimeter × apothem

    Half the perimeter times the apothem.

    A=12×24×3.4641016≈41.569219A = \tfrac{1}{2} \times 24 \times 3.4641016 \approx 41.569219
Formulas used
Interior angle=(n−2) 180∘n\text{Interior angle} = \frac{(n-2)\,180^\circ}{n}
A=ns24tan⁡(π/n)A = \frac{n s^2}{4\tan(\pi/n)}
How this was checked
  • ✓ Two area formulas agree, and each interior and exterior angle add to 180°.

Frequently asked questions

What is the interior angle of a hexagon?

(6 − 2) × 180° ÷ 6 = 120°.

What do the interior angles of a polygon add up to?

(n − 2) × 180°. For a pentagon that is 540°.

What is the apothem?

The distance from the centre to the middle of a side. Area = ½ × perimeter × apothem.

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