Walk all the way round any polygon and you turn through exactly one full turn. That single fact gives the exterior angles, and the interior angles follow.
The exterior angles of any polygon add up to \(360^\circ\), and at each vertex interior + exterior \(= 180^\circ\). The interior angles of an \(n\)-sided polygon add up to \((n - 2) \times 180^\circ\). For a regular polygon, each exterior angle is \(\dfrac{360^\circ}{n}\).
Fig. 1: a regular pentagon drawn to scale. Exterior angle = 360° ÷ 5 = 72°, so interior angle = 180° − 72° = 108°
Section 01
The angle sum.
In the polygons tab, change the number of sides and watch the polygon split into triangles from one vertex. Count the triangles each time: it is always two fewer than the number of sides.
Polygon
Sides \(n\)
Triangles \(n - 2\)
Angle sum
Triangle
3
1
\(180^\circ\)
Quadrilateral
4
2
\(360^\circ\)
Pentagon
5
3
\(540^\circ\)
Hexagon
6
4
\(720^\circ\)
Octagon
8
6
\(1080^\circ\)
Decagon
10
8
\(1440^\circ\)
Worked example A missing angle (Extended: irregular polygon)
Four of the interior angles of a pentagon are \(100^\circ\), \(120^\circ\), \(95^\circ\) and \(110^\circ\). Find the fifth angle.
Angle sum of a pentagon \(= (5 - 2) \times 180^\circ = 540^\circ\).
The four known angles total \(100 + 120 + 95 + 110 = 425^\circ\).
The interior angles of a polygon add up to \(1440^\circ\). How many sides does it have?
\((n - 2) \times 180 = 1440\).
\(n - 2 = 1440 \div 180 = 8\).
\(n = 10\).
Answer: 10 sides (a decagon)
Examiner trap
The formula is \((n - 2) \times 180^\circ\), not \(n \times 180^\circ\). Splitting from one vertex gives \(n - 2\) triangles; a pentagon makes 3 triangles, not 5.
Section 02
Exterior angles and regular polygons.
Imagine walking round the pentagon in Fig. 1. At each corner you turn through the exterior angle, and after five turns you face the way you started: one full turn, \(360^\circ\).
Any polygon: the exterior angles add up to \(360^\circ\), and interior + exterior \(= 180^\circ\) at each vertex (angles on a straight line).
Can a regular polygon have interior angles of \(130^\circ\)? Explain.
Exterior angle would be \(180^\circ - 130^\circ = 50^\circ\), and \(360 \div 50 = 7.2\). The number of sides must be a whole number, so no.
Examiner trap
Dividing 360 by the INTERIOR angle is a common slip: \(360 \div 156\) is not a number of sides. Convert to the exterior angle first.
Stage 1 · Learn
Check your understanding
Four quick checks. Each correct answer earns XP and records this skill on your profile.
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What do the exterior angles of any polygon add up to?
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What is the sum of the interior angles of a hexagon?
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What is the exterior angle of a regular octagon?
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At each vertex of a polygon, the interior and exterior angles add up to
Section 03
Polygons together.
Questions often fit polygons together at a point or along an edge. Work out each polygon's interior angle, then use angles at a point (\(360^\circ\)) or on a straight line (\(180^\circ\)).
Worked example A square and a hexagon
A regular hexagon and a square share a side, and lie on opposite sides of it. Find the angle between them at a shared vertex, outside both shapes.