Regular Polygon Calculator

Q: How do you find one interior angle of a regular polygon?

Because all angles are equal, divide the total by the number of sides: (n minus 2) times 180 degrees divided by n. A regular pentagon gives 108 degrees and a regular octagon gives 135 degrees.

Q: What is the exterior angle of a regular polygon?

It is 360 degrees divided by n, since the exterior angles of any polygon sum to 360 degrees. The interior and exterior angle at each corner add to 180 degrees.

Q: What is the difference between the circumradius and the apothem?

Both start at the centre, but the circumradius reaches a vertex while the apothem reaches the midpoint of a side. The circumradius is always the larger of the two.

Q: How do you find the area of a regular polygon?

From the side length: A equals one quarter times n times s squared times the cotangent of 180 degrees over n. Or, with the apothem known, area equals one half times perimeter times apothem. For a hexagon of side 10 that is about 259.81.

Choose the number of sides, then enter any one size — side length, circumradius, apothem or area — and the rest follow, including the interior and exterior angles. Click a value on the drawing to change it.

Polygon

Hexagon

Number of sides

Sides (n)

Enter any one size

Side length (s)

Circumradius (R)

Apothem (a)

Area

Units Decimals

Click an amber value to edit it · not to printed scale

Results

Regular polygon formulas

For a regular polygon with n sides of length s, every measurement comes from n and one size value.

Area · A = ¼ · n · s² · cot(π/n)

Circumradius · R = s / (2 sin(π/n))

Apothem · a = s / (2 tan(π/n))

Angles

Each interior angle is (n − 2) × 180° ÷ n, and each exterior angle is 360° ÷ n. The interior and exterior angles at any vertex always add to 180°.

Terms

The circumradius reaches from the centre to a vertex; the apothem (inradius) reaches from the centre to the middle of a side. The perimeter is simply n times the side length.

Calculating a regular polygon

A regular polygon has every side the same length and every angle the same size — like an equilateral triangle, a square, a regular pentagon or hexagon, and so on. That regularity means the whole shape is pinned down by just two things: the number of sides, and any one size measurement. The results split neatly into two groups: the angles, which depend only on how many sides there are, and the sizes, which scale with the side length.

Angles depend only on the number of sides

Sum of interior angles = (n − 2) × 180°. A four-sided shape totals 360°, a hexagon 720°, and so on.
Each interior angle = (n − 2) × 180° ÷ n, since all the angles are equal.
Each exterior angle = 360° ÷ n. The exterior angles of any polygon always add up to 360°.
At every corner the interior and exterior angle add to 180°, because they sit on a straight line.

Sizes: side, radius, apothem and area

Circumradius (R) reaches from the centre out to a vertex (corner).
Apothem (a) reaches from the centre to the midpoint of a side — it's the radius of the inscribed circle.
Perimeter is just n times the side length.
Area can be found from the side directly, or as half the perimeter times the apothem.

Worked example: a regular hexagon

Take a regular hexagon — n = 6 sides — with side length s = 10.

Step 1 — interior and exterior angles

interior = (6 − 2)×180° ÷ 6 = 720° ÷ 6 = 120° · exterior = 360° ÷ 6 = 60°

Step 2 — perimeter

P = n × s = 6 × 10 = 60

Step 3 — circumradius (centre to a corner)

R = s ÷ (2·sin(180°/n)) = 10 ÷ (2·sin 30°) = 10 ÷ 1 = 10

Step 4 — apothem (centre to a side's midpoint)

a = s ÷ (2·tan(180°/n)) = 10 ÷ (2·tan 30°) = 8.6603

Step 5 — area, as half the perimeter times the apothem

A = ½ · P · a = ½ · 60 · 8.6603 = 259.81

Result

Interior angle 120°, exterior 60°, perimeter 60, circumradius 10, apothem 8.66, area 259.81. A hexagon is a special case where the circumradius exactly equals the side length.

Formulas this calculator uses

Interior angle · (n − 2)·180° ÷ n

Exterior angle · 360° ÷ n

Area · ¼·n·s²·cot(π/n) = ½·P·a

Circumradius · R = s ÷ (2·sin(π/n))

Apothem · a = s ÷ (2·tan(π/n))

Frequently asked questions

What is the sum of the interior angles of a polygon?

For any polygon with n sides, the interior angles add up to (n − 2) × 180°. A triangle totals 180°, a quadrilateral 360°, a pentagon 540°, and a hexagon 720°. This holds for irregular polygons too, not just regular ones.

How do you find one interior angle of a regular polygon?

Because all the angles are equal, divide the total by the number of sides: (n − 2) × 180° ÷ n. A regular pentagon gives 540° ÷ 5 = 108°, and a regular octagon gives 1080° ÷ 8 = 135°.

What is the exterior angle of a regular polygon?

It's 360° ÷ n, since the exterior angles of any polygon always sum to a full turn of 360°. The interior and exterior angle at each corner add to 180°, so you can find either one from the other.

What is the apothem of a polygon?

The apothem is the distance from the centre to the midpoint of a side — the radius of the circle that fits snugly inside. It's given by a = s ÷ (2·tan(180°/n)), and it makes the area easy: area equals half the perimeter times the apothem.

What is the difference between the circumradius and the apothem?

Both start at the centre, but the circumradius reaches out to a vertex (a corner), while the apothem reaches only to the middle of a side. The circumradius is therefore always the larger of the two.

How do you find the area of a regular polygon?

From the side length: A = ¼·n·s²·cot(180°/n). Or, if you already have the apothem, the simpler route is A = ½ × perimeter × apothem. Both give the same answer — for a hexagon of side 10, about 259.81 square units.