Regular Polygon Calculator: Interior and Exterior Angles, Side, Apothem, Area and Diagonals

Get every dimension and angle of a regular polygon from one measurement.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Enter the number of sides, then choose whether you know the side, the circumradius or the apothem, and enter that length.

The calculator gives the angle sum (n - 2) x 180, each interior angle, each exterior angle 360 over n, the side, circumradius, apothem, widths across flats and corners, perimeter, area and diagonals.

For a hexagon it also tells you the across flats and across corners sizes used for bar and nuts.

Formulas used

  • \sum\text{interior} = (n - 2)\times180^\circ
  • \text{each exterior} = \frac{360^\circ}{n}
  • s = 2R\sin\frac{\pi}{n}
  • a = \frac{s}{2\tan(\pi/n)}
  • A = \tfrac12 n s a

Worked example

With the default numbers in the calculator above:

  1. Sum of the interior angles: 720.00 degrees
  2. Each interior angle of the regular polygon: 120.00 degrees
  3. Each exterior angle (all the exterior angles of any convex polygon add up to 360 degrees): 60.00 degrees
  4. Length of one side: 50.00
  5. Circumradius (centre to a corner): 50.00
  6. Apothem (centre to the middle of a side): 43.30
  7. Width across flats (twice the apothem) and across corners (twice the circumradius, for an even number of sides): 86.60 across flats and 100.00 across corners
  8. Perimeter: 300.00
  9. Area: 6495.19 square units
  10. Number of diagonals, and of triangles it can be cut into: 9 diagonals and 4 triangles

Change the numbers in the calculator to see the full working, every step, for your own case.

Common questions

Why is the angle sum (n - 2) x 180?

A polygon can be cut from one corner into n - 2 triangles, and each triangle has 180 degrees.

What do the exterior angles add up to?

360 degrees for any convex polygon.

What is across flats?

The distance between two opposite sides, which is the spanner size of a hexagon nut. It is twice the apothem.

How many diagonals does a polygon have?

n (n - 3) over 2. A hexagon has 9.

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Any school or university mathematics text (polygons); ISO 272 Fasteners: hexagon products, widths across flats.

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