Polygon angle sums

Triangulate from one vertex

A convex polygon with n sides can be split into n-2 non-overlapping triangles by diagonals from one vertex. Its interior-angle sum is therefore (n-2)180°.

A hexagon’s total is 4 × 180° = 720°. This does not make every interior angle 120° unless the angles are equal, as in a regular hexagon. A regular polygon has equal sides and equal interior angles.

One complete exterior turn

Walking once around a convex polygon, the exterior turns in the same direction total 360°. In a regular n-gon, each exterior angle is 360°/n and each interior angle is 180°-360°/n.

Regular polygon	Interior angle	Exterior angle	Rotational order
Pentagon	108°	72°	5
Hexagon	120°	60°	6
Octagon	135°	45°	8
Decagon	144°	36°	10

Each also has n lines of symmetry. A non-regular polygon with the same side count need not share these individual angles or symmetries.

Check your understanding 1 Reverse the exterior angle

A regular polygon has exterior angle 24°. Find its number of sides and one interior angle.

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It has 360/24 = 15 sides. One interior angle is 180° − 24° = 156°. Dividing the interior-angle sum by 24° would mix a total with an exterior angle.

Check an impossible claim

Can a regular polygon have an exterior angle of 50°?

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No. The required side count would be 360/50 = 7.2, which is not an integer. A calculation must satisfy the geometric domain, not merely produce a number.

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