Interior and Exterior Angles of Polygons
This page shows you how to work out the interior and exterior angles of polygons for IGCSE Maths (Pearson Edexcel 4MA1). You will learn the angle sum rule for any polygon, the shortcuts for a regular polygon, and how to find the number of sides when an angle is given. Worked examples and free, auto-marked practice questions are included so you can test yourself straight away.
What is a polygon?
A polygon is a closed shape with three defining properties:
- It is two-dimensional (2D). A polygon is a flat shape, not a solid.
- It is made of straight lines. Every side is a straight edge: no curves are allowed.
- The lines are connected. The sides join end to end to enclose the shape, with no gaps and no loose ends.
A polygon is regular when all of its sides are equal in length and all of its angles are equal in size. A square is a regular polygon; a rectangle that is not a square is not, because its sides are not all equal.
Interior and exterior angles
At every corner the interior angle (inside) and the exterior angle (outside, where a side is extended) lie on a straight line, so they always add up to 180°.
Regular polygons: a faster route
For a regular polygon it is usually quickest to find the exterior angle first, because it is a single division: \( \dfrac{360^\circ}{n} \). Once you have the exterior angle, the interior angle is just \( 180^\circ \) minus it. For example, a regular pentagon has an exterior angle of \( \dfrac{360^\circ}{5}=72^\circ \), so each interior angle is \( 180^\circ-72^\circ=108^\circ \).
Why do the exterior angles add up to 360°?
Imagine walking once around the outside of a polygon. At each corner you turn through the exterior angle, and by the time you get back to the start you have turned through one full circle. That is why the exterior angles of any polygon, regular or not, always total \(360^\circ\).
Use the demonstration below. Slide the control to gather the exterior angles into the centre: they fit together exactly to make a full circle. Change the number of sides to check that this works for every polygon.
Worked examples
💡 Example 1
Work out the sum of the interior angles of a polygon with 9 sides.
\[ (n-2)\times 180^\circ \]
\[ (9-2)\times 180^\circ \]
\[ 7\times 180^\circ = 1260^\circ \]
What is happening?
Use the angle sum rule with \(n=9\).
A 9-sided polygon splits into 7 triangles.
💡 Example 2
A regular polygon has 10 sides. Work out the size of each interior angle.
\[ \text{exterior} = \frac{360^\circ}{10} = 36^\circ \]
\[ \text{interior} = 180^\circ - 36^\circ \]
\[ = 144^\circ \]
What is happening?
Find the exterior angle first, as one quick division.
Interior and exterior sit on a straight line.
💡 Example 3
Each exterior angle of a regular polygon is 24°. How many sides does it have?
\[ n = \frac{360^\circ}{\text{exterior}} \]
\[ n = \frac{360^\circ}{24^\circ} \]
\[ n = 15 \]
What is happening?
The exterior angles total \(360^\circ\), shared equally.
So divide \(360^\circ\) by one exterior angle.
💡 Example 4
A pentagon has interior angles of \(2x\), \(2x\), \((x+40)^\circ\), \((x+40)^\circ\) and \(100^\circ\). Find \(x\).
\[ \text{sum} = (5-2)\times 180^\circ = 540^\circ \]
\[ 2x+2x+(x+40)+(x+40)+100 = 540 \]
\[ 6x+180 = 540 \]
\[ 6x = 360 \]
\[ x = 60 \]
What is happening?
The pentagon angle sum is \(540^\circ\).
Add all five expressions and set the total equal to \(540\).
Then solve the linear equation.
🔑 Key points
- The interior angle sum is \((n-2)\times 180^\circ\) for any polygon.
- The exterior angles always add up to \(360^\circ\).
- For a regular polygon, the exterior angle is the quick one: \( \frac{360^\circ}{n} \).
- Interior angle plus exterior angle equals \(180^\circ\) at every vertex.
⚠️ Common pitfalls
- Using \(n\times 180^\circ\) instead of \((n-2)\times 180^\circ\) for the sum.
- Dividing by \(n\) to find each angle when the polygon is not regular.
- Mixing up interior and exterior, then subtracting from the wrong total.
- Forgetting that the exterior sum is \(360^\circ\), not \(180^\circ\).