Angle Sums in Polygons Revision Notes | GCSE AQA Higher Maths

1. Why Triangles Are the Key to Every Polygon

You already know that the angles in a triangle sum to $180^\circ$. That single fact is the engine behind every angle-sum result in polygons — from quadrilaterals to dodecagons. The question is: how do we get from triangles to everything else?

The answer is diagonal triangulation. Pick any vertex of a polygon and draw straight lines (diagonals) to every non-adjacent vertex. The polygon splits into triangles, and since each triangle contributes $180^\circ$, you can count the triangles to find the total angle sum.

A pentagon splits into 3 triangles, giving $3 \times 180^\circ = 540^\circ$. A hexagon splits into 4 triangles: $4 \times 180^\circ = 720^\circ$. The pattern is clear — a polygon with $n$ sides always splits into $(n - 2)$ triangles.

Notice how each extra side adds exactly $180^\circ$ to the total. That is not a coincidence — each new side creates one more triangle in the triangulation. This table is worth memorising for quick reference in questions without a calculator.

2. Finding Missing Angles in Polygons

Now that you have the formula, you can use it to find a missing angle when all the others are given. The method is always the same: find the total, subtract the known angles, and what remains is the missing one.

Worked example: Finding a Missing Interior Angle in a Hexagon

A hexagon has five interior angles of $110^\circ$, $125^\circ$, $130^\circ$, $95^\circ$, and $118^\circ$. Find the sixth angle.

1. $(n - 2) \times 180^\circ = (6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ$ — Substitute $n = 6$ into the formula. [M1] for correct use of formula.
2. $110 + 125 + 130 + 95 + 118 = 578^\circ$ — Add carefully: $110 + 125 = 235$, $235 + 130 = 365$, $365 + 95 = 460$, $460 + 118 = 578$.
3. $720^\circ - 578^\circ = 142^\circ$ — [A1] for the correct final answer.
4. $578 + 142 = 720$ ✓ — Confirm the six angles sum to $720^\circ$ as required.

Answer: The sixth angle is $142^\circ$.

3. Exterior Angles and Their Surprising Sum

At each vertex of a polygon, the interior angle and the exterior angle sit on a straight line, so they sum to $180^\circ$. The exterior angle is formed by extending one side of the polygon outward.

Here is the result that surprises most students: no matter how many sides a polygon has, its exterior angles always add up to exactly $360^\circ$. Imagine walking around the outside of any polygon — you turn through a total of one full rotation, which is $360^\circ$.

This result is powerful because it gives you a second route to finding unknown angles. If you know all exterior angles but one, subtract their sum from $360^\circ$. You can also convert between interior and exterior angles using the straight-line relationship: exterior angle $= 180^\circ -$ interior angle.

4. Regular Polygons — Equal Angles, Elegant Results

A regular polygon has all sides equal and all interior angles equal. That symmetry makes calculations much cleaner: once you know the angle sum, divide by the number of sides to find each individual angle.

Notice that the exterior angle formula $\frac{360^\circ}{n}$ is often the faster approach for regular polygons. Find the exterior angle first, then subtract from $180^\circ$ to get the interior angle — it avoids multiplying and dividing large numbers.

Worked example: Finding Angles in a Regular Octagon

Find the interior angle and exterior angle of a regular octagon.

1. $\text{Exterior angle} = \dfrac{360^\circ}{8} = 45^\circ$ — For a regular polygon, divide $360^\circ$ by the number of sides. [M1]
2. $\text{Interior angle} = 180^\circ - 45^\circ = 135^\circ$ — Interior and exterior angles are supplementary (sum to $180^\circ$). [A1]
3. $(8 - 2) \times 180^\circ = 6 \times 180^\circ = 1080^\circ$ $1080^\circ \div 8 = 135^\circ$ ✓ — Both methods agree, confirming the answer is correct.

Answer: Exterior angle $= 45^\circ$; Interior angle $= 135^\circ$.

You can also work backwards: if you are told the interior or exterior angle of a regular polygon, you can find the number of sides. Divide $360^\circ$ by the exterior angle to get $n$. This is a favourite exam question type.

Worked example: Finding the Number of Sides from an Exterior Angle

A regular polygon has an exterior angle of $24^\circ$. How many sides does it have?

1. $n = \dfrac{360^\circ}{\text{exterior angle}} = \dfrac{360^\circ}{24^\circ}$ — Rearrange $\text{exterior angle} = \frac{360^\circ}{n}$ to make $n$ the subject. [M1]
2. $n = \dfrac{360}{24} = 15$ — $360 \div 24 = 15$. [A1] The answer must be a whole number — if it is not, recheck your working.

Answer: The polygon has 15 sides.

5. Deliberate Practice

These problems cover the full range of question types you will meet in an Higher-tier exam. Work through each one before checking the solution — the method marks come from showing clear, structured working.

6. Exam Technique and Common Mistakes

Angle questions In exam questions almost always require you to state a reason alongside your calculation. Writing the angle value alone, without the geometric justification, will cost you a mark. Practise writing concise reasons such as "interior angle sum of a hexagon is $720^\circ$" or "exterior angles of a polygon sum to $360^\circ$".

When a question says "show that" or "prove", you must start from a known result — usually the triangle angle sum — and build logically to the target. Never work backwards from the answer you are trying to prove.