Angles in Polygons

A polygon is a flat shape with straight sides. This page is about the angles in polygons: the total of the angles inside, the size of one angle, and how to find the number of sides from an angle.

You need these names:

• Pentagon: 5 sides
• Hexagon: 6 sides
• Octagon: 8 sides

In the exam you are usually asked to work out the sum of the interior angles, find a missing angle, or find an angle of a regular polygon, for 2 or 3 marks. Harder questions, for 4 or 5 marks, join these facts together: shapes meeting at a point, or angles described in words, such as “angle D is twice angle C”.

An interior angle is an angle inside the polygon, at one of its corners. An exterior angle is outside: extend one side in a straight line, and the exterior angle is between that line and the next side.

Angles on a straight line add up to 180^\circ, so at every corner:

\text{interior angle} + \text{exterior angle} = 180^\circ

A regular polygon has all its sides equal and all its angles equal. If a question does not say the polygon is regular, do not assume the angles are equal, even if they look it.

The angles in each triangle add up to 180^\circ. The three triangles fill the pentagon, so the interior angles of a pentagon add up to 3 \times 180^\circ = 540^\circ

A polygon with n sides always splits into n - 2 triangles in this way. So:

\text{sum of interior angles} = (n - 2) \times 180^\circ

where n is the number of sides. This is the total of all the interior angles, not the size of one angle.

• Quadrilateral: 2 \times 180^\circ = 360^\circ
• Pentagon: 3 \times 180^\circ = 540^\circ
• Hexagon: 4 \times 180^\circ = 720^\circ
• Octagon: 6 \times 180^\circ = 1080^\circ

Imagine walking all the way round the edge of a polygon. At each corner you turn through the exterior angle. When you get back to the start, you are facing the same way again, so you have made one full turn.

So the exterior angles of a polygon add up to 360^\circ. In a regular polygon with n sides, the exterior angles are all equal, so:

\text{exterior angle} = 360^\circ \div n

Then the interior angle is 180^\circ minus the exterior angle.

In a regular polygon, the exterior angles share the full turn equally. So:

n = 360^\circ \div \text{exterior angle}

For example, an exterior angle of 45^\circ gives 360 \div 45 = 8 sides.

If you are given the interior angle, take it away from 180^\circ first to get the exterior angle. Then divide 360^\circ by the exterior angle. Never divide 360 by the interior angle.

Sometimes the interior angle is described using the exterior angle. If the interior angle is 2 times the exterior angle, the two angles make 2 + 1 = 3 equal parts of 180^\circ. So the exterior angle is 180^\circ \div 3 = 60^\circ, and the polygon has 360 \div 60 = 6 sides.

The number of sides must be a whole number. If you get a decimal, check your working.

In harder questions, some angles are described in words, such as “angle F is twice angle E”. Call the smaller unknown angle x, write the other angles using x, and use the angle sum to make an equation.

When shapes meet at a point with no gaps and no overlaps, their angles at that point add up to 360^\circ. Work out each shape's interior angle first, then take them away from 360^\circ to find what is left.

Write each angle you find on the diagram, at the right corner. Do not measure the diagram: it is usually not drawn to scale.

• Mixing up interior and exterior angles. In a regular polygon, 360^\circ \div n is the exterior angle; take it from 180^\circ to get the interior angle.
• Using 360^\circ as the interior angle sum of every polygon. Only a quadrilateral's angles add up to 360^\circ, so use (n - 2) \times 180^\circ for other polygons.
• Sharing the total equally when the angles are not equal. Divide the total by the number of angles only when the polygon is regular.
• Dividing 360 by the interior angle. Find the exterior angle first, then divide 360^\circ by it to get the number of sides.
• Guessing angles from the diagram. Diagrams are not drawn to scale, so do not measure them or assume angles look equal.
• Slips when dividing. Check your answer is sensible: 540 \div 5 = 108, not 18, because 5 \times 18 is only 90.