Sum of interior angles, each interior and exterior angle of a regular polygon and the number of diagonals, from the number of sides or one angle.
By AdminDrafted with AI writing toolsUpdated Checked examples: 8
A Pentagon (5 sides) has interior angles adding to 540°; when regular, each is 108° and each exterior angle is 72°. It has 5 diagonals.
A 5-sided convex polygon splits into 3 triangles from one vertex.
A simple polygon with n sides can be split into n − 2 triangles, so its interior angles add up to (n − 2) × 180°. In a regular polygon all the angles are equal: each interior angle is (n − 2) × 180°/n and each exterior angle 360°/n, because the exterior angles of a convex polygon make one full turn. The number of diagonals is n(n − 3)/2.
The default pentagon has interior angles adding up to 540°, or 108° each when regular, with 72° exterior angles and 5 diagonals. Working backwards is common in tiling and woodwork: an interior angle of 140° means 360/(180 − 140) = 9 sides, and each joint of a regular n-sided frame is mitred at 180°/n, 36° for a pentagon.
An angle that fits no whole number of sides, such as 100°, is rejected with the nearest valid polygon. A rounded entry is accepted when it matches an exact angle: 128.57° is read as the heptagon's 900/7°.
The interior angles of any simple polygon with n sides add up to (n − 2) × 180°, which follows from the triangle angle sum in Euclid, Elements I.32. A hexagon's total is 4 × 180° = 720°, so each angle of a regular hexagon is 120°. The total applies to irregular polygons too; the equal share only to regular ones.
Diagonals drawn from one corner of a convex polygon cut it into n − 2 triangles, and each triangle's angles add to 180°. A pentagon splits into 3 triangles, so its angles total 540°.
Any simple polygon, convex or concave, can be cut into n − 2 triangles, so the rule applies whenever the sides do not cross. In radians the sum is (n − 2)π, which is 10π ≈ 31.4159 rad for a 12-sided dodecagon.
In a regular polygon every side and every angle is equal:
An interior angle and the exterior angle beside it always add to 180° because they lie along one straight line.
| Sides | Interior-angle sum | Each interior angle | Each exterior angle | Diagonals |
|---|---|---|---|---|
| 3 | 180° | 60° | 120° | 0 |
| 4 | 360° | 90° | 90° | 2 |
| 5 | 540° | 108° | 72° | 5 |
| 6 | 720° | 120° | 60° | 9 |
| 8 | 1080° | 135° | 45° | 20 |
| 12 | 1800° | 150° | 30° | 54 |
As n grows the interior angle approaches 180°: a 1,000-sided polygon has angles of 179.64°.
Divide 360° by the exterior angle: n = 360° ÷ (180° − interior angle). An interior angle of 140° gives 360 ÷ 40 = 9 sides, a nonagon with 1,260° of angles in total.
Starting from an exterior angle of 24° gives 360 ÷ 24 = 15 sides, each with a 156° interior angle.
When n is not a whole number, no regular polygon has that angle. An interior angle of 100° gives 360 ÷ 80 = 4.5 sides; the nearest regular polygon, the pentagon, has 108°.
Some regular angles repeat forever as decimals. A heptagon's interior angle is 900/7 ≈ 128.5714°, so a rounded 128.57° still identifies 7 sides.
A polygon with n vertices has n(n − 3)/2 diagonals. Each vertex connects to all the others except itself and its two neighbors, n − 3 in all; dividing by 2 stops each diagonal being counted twice. An octagon has 8 × 5 ÷ 2 = 20.
To build a regular frame, planter or gazebo from straight pieces, cut both ends of each piece at 180°/n from square. That is 36° for a pentagon, 30° for a hexagon and 22.5° for an octagon. Two such cuts meet at the polygon's interior angle.
For the area of a regular polygon from its side length, use the area and perimeter calculator. For an irregular outline, use polygon area from coordinates.
Checked against: Angle sum of a triangle (Euclid I.32); a triangle has no diagonals
Checked against: 3 × 180; 540/5; 360/5; 5·2/2
Checked against: 900/7, 360/7; 7·4/2
Checked against: 10 × 180; 1800/12; 12·9/2
(n − 2) × 180°, where n is the number of sides. A triangle has 180°, a quadrilateral 360°, a pentagon 540°, a hexagon 720° and an octagon 1,080°. The rule holds for any simple polygon, regular or not, because it can be triangulated into n − 2 triangles of 180° each.
Divide the angle sum by the number of sides: (n − 2) × 180°/n, which is the same as 180° − 360°/n. A regular hexagon has 120° angles, an octagon 135° and a dodecagon 150°. The angle approaches 180° as n grows; a 1,000-sided polygon has angles of 179.64°.
Subtract the angle from 180° to get the exterior angle, then divide 360° by it: n = 360°/(180° − interior angle). An interior angle of 140° gives 360/40 = 9 sides, a nonagon. If the result is not a whole number, no regular polygon has that angle: 100° gives 4.5.
360° for any convex polygon, whatever the number of sides. Walking once around the boundary, you turn through each exterior angle and finish facing your starting direction, which is one full turn. In a regular polygon each exterior angle is 360°/n: 72° for a pentagon, 60° for a hexagon and 45° for an octagon.
n(n − 3)/2. Each of the n vertices joins n − 3 others by a diagonal (not itself and not its two neighbours), and every diagonal is counted from both ends, hence the division by 2. A pentagon has 5 diagonals, a hexagon 9, an octagon 20 and a dodecagon 54; a triangle has none.
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