What is the formula for the interior angle sum of a polygon?

For a polygon with $n$ sides, the interior angle sum is $(n-2) \times 180^\circ$. The formula works because the polygon can be divided into $n-2$ triangles.

How do you find one interior angle of a regular polygon?

Calculate the total interior angle sum using $(n-2) \times 180^\circ$, then divide by $n$. Alternatively, calculate one exterior angle using $360^\circ \div n$ and subtract it from $180^\circ$.

Do exterior angles always add up to 360 degrees?

Yes, one exterior angle at each vertex adds to $360^\circ$ when the angles are measured in the same direction around a standard polygon. The exterior angles are equal only if the polygon is regular.

How do you find the number of sides from an exterior angle?

For a regular polygon, divide $360^\circ$ by the size of one exterior angle: $n=360^\circ \div \text{exterior angle}$. The result must be a whole number.

How do you find the number of sides from an interior angle?

For a regular polygon, subtract the interior angle from $180^\circ$ to find the exterior angle. Then divide $360^\circ$ by that exterior angle.

What does regular polygon mean in GCSE Maths?

A regular polygon has all sides equal in length and all interior angles equal in size. You should not assume that a polygon is regular unless the question states this or marks it clearly.

Angles in Polygons GCSE Maths: Revision and Questions

Angles in polygons GCSE Maths questions require you to use a small number of connected angle facts. The key results are that the sum of the interior angles of an $n$ -sided polygon is $\times 180^\circ$ , while the exterior angles of any polygon add to $360∘360^\circ$ .

This guide explains where the rules come from, how to select the correct formula and how to show enough working for an exam. It includes angles in polygons worked examples, common mistakes and exam-style questions with answers.

What is a polygon?

A polygon is a closed, two-dimensional shape made from straight sides. The points where two sides meet are called vertices.

A regular polygon has sides of equal length and interior angles of equal size. If either the sides or the angles are not all equal, the polygon is irregular.

Common polygon names include:

Triangle: $3$ sides
Quadrilateral: $4$ sides
Pentagon: $5$ sides
Hexagon: $6$ sides
Heptagon: $7$ sides
Octagon: $8$ sides
Nonagon: $9$ sides
Decagon: $10$ sides

The word regular is important. You can divide an angle sum by the number of sides to find each interior angle only when the polygon is regular.

Interior and exterior angles

An interior angle lies inside a polygon at a vertex. An exterior angle is formed between one side and the extension of an adjacent side.

At each vertex, the interior angle and the corresponding exterior angle form a straight line. Therefore:

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

This relationship is useful when you know one type of angle but need the other.

The sum of the interior angles of a polygon

For a polygon with $n$ sides, the interior angle sum is:

\times 180^\circ

This formula works because a polygon can be divided into triangles by drawing diagonals from one vertex. An $n$ -sided polygon produces $n - 2$ triangles, and each triangle has an angle sum of $180∘180^\circ$ .

For example, a pentagon can be divided into $3$ triangles, so its interior angle sum is:

\times 180^\circ=540^\circ

The formula gives the total, not the size of each angle.

Number of sides	Polygon	Interior angle sum
$3$	Triangle	$180∘180^\circ$
$4$	Quadrilateral	$360∘360^\circ$
$5$	Pentagon	$540∘540^\circ$
$6$	Hexagon	$720∘720^\circ$
$8$	Octagon	$1080∘1080^\circ$
$10$	Decagon	$1440∘1440^\circ$

Worked example 1: finding a missing interior angle

An irregular hexagon has five interior angles of $105∘105^\circ$ , $112∘112^\circ$ , $126∘126^\circ$ , $130∘130^\circ$ and $107∘107^\circ$ . Find the sixth angle, $x$ .

A hexagon has $6$ sides, so its interior angle sum is:

\times 180^\circ=720^\circ

Add the five known angles:

105∘+112∘+126∘+130∘+107∘=580∘105^\circ+112^\circ+126^\circ+130^\circ+107^\circ=580^\circ

Subtract this from the total:

x=720∘−580∘=140∘x=720^\circ-580^\circ=140^\circ

The missing angle is $140∘140^\circ$ . Notice that the angles were not divided by $6$ because the hexagon was irregular.

Angles in regular polygons

All the interior angles of a regular polygon are equal. To find one interior angle, divide the total interior angle sum by the number of sides:

angle=(n−2)×180∘n\text{one interior angle}=\frac{(n-2) \times 180^\circ}{n}

The exterior angles of any polygon add to $360∘360^\circ$ . In a regular polygon, the exterior angles are equal, so:

angle=360∘n\text{one exterior angle}=\frac{360^\circ}{n}

You can then find an interior angle using:

angle\text{one interior angle}=180^\circ-\text{one exterior angle}

The exterior-angle method is often the quickest approach for regular polygons.

Worked example 2: interior angle of a regular nonagon

A regular nonagon has $9$ sides. Its exterior angles are equal, so one exterior angle is:

360∘÷9=40∘360^\circ \div 9=40^\circ

The interior and exterior angles lie on a straight line:

180∘−40∘=140∘180^\circ-40^\circ=140^\circ

Each interior angle is $140∘140^\circ$ .

You could instead use the interior-angle formula:

(9−2)×180∘9=140∘\frac{(9-2) \times 180^\circ}{9}=140^\circ

Both methods are correct, but the exterior-angle method involves fewer steps.

Worked example 3: finding the number of sides

Each interior angle of a regular polygon is $165∘165^\circ$ . Find the number of sides.

First find the exterior angle:

180∘−165∘=15∘180^\circ-165^\circ=15^\circ

The exterior angles add to $360∘360^\circ$ , so the number of equal exterior angles is:

n=360∘÷15∘=24n=360^\circ \div 15^\circ=24

The polygon has $24$ sides.

The answer must be a whole number. A result such as $12.5$ sides would indicate an arithmetic error or that the stated angle cannot belong to a regular polygon.

Algebra in angles in polygons questions

Some exam questions describe interior angles using algebra. The method is still based on finding the correct angle sum, forming an equation and solving it.

Worked example 4: forming an equation

The interior angles of a quadrilateral are $x$ , $x + 10$ , $2 x$ and $2 x + 20$ . Find $x$ .

A quadrilateral has an interior angle sum of $360∘360^\circ$ , so:

x + (x + 10) + 2 x + (2 x + 20) = 360

Collect like terms:

6 x + 30 = 360

6 x = 330

x = 55

The four angles are $55∘55^\circ$ , $65∘65^\circ$ , $110∘110^\circ$ and $130∘130^\circ$ . Checking them gives:

55∘+65∘+110∘+130∘=360∘55^\circ+65^\circ+110^\circ+130^\circ=360^\circ

The check confirms that the values satisfy the angle sum.

Choosing the correct method

Information given	Recommended method
Number of sides and several interior angles	Find $\times 180^\circ$ , then subtract known angles
Regular polygon and number of sides	Use $360∘÷n360^\circ \div n$ for one exterior angle
One exterior angle of a regular polygon	Use $angle360^\circ \div \text{exterior angle}$
One interior angle of a regular polygon	Subtract from $180∘180^\circ$ , then use the exterior-angle rule
Total interior angle sum	Solve $\times 180^\circ=\text{given sum}$

Writing the formula before substituting values makes your method clear. This can earn method marks even if a later arithmetic error leads to the wrong final answer.

For a guided explanation and further demonstrations, use the MathsGenie angles in polygons videos and the main angles in polygons revision page.

Common mistakes in angles in polygons GCSE Maths

Confusing the angle sum with one angle

For a regular hexagon, $\times 180^\circ=720^\circ$ is the total interior angle sum. Each interior angle is:

720∘÷6=120∘720^\circ \div 6=120^\circ

Dividing by the number of sides for an irregular polygon

Equal division is valid only when the polygon is regular. For an irregular polygon, use the total angle sum and subtract the angles you know.

Using $\times 180^\circ$

The polygon must be divided into $n - 2$ triangles, not $n$ triangles. The correct interior sum is $\times 180^\circ$ .

Forgetting that interior and exterior angles total $180∘180^\circ$

They form a straight line, so they are supplementary. They do not normally add to $360∘360^\circ$ at one vertex.

Assuming the exterior-angle rule applies only to regular polygons

The exterior angles add to $360∘360^\circ$ for any standard polygon when one exterior angle is taken at each vertex in the same direction. Regularity is needed only when dividing $360∘360^\circ$ equally to find one exterior angle.

Giving a decimal number of sides

The number of sides must be a positive whole number. Check your subtraction and division if your answer is not an integer.

Angles in polygons exam questions

Attempt each question before reading the answers. Show the formula or angle fact used, not just the final number.

Question 1

Find each interior angle of a regular $12$ -sided polygon.

Question 2

Each exterior angle of a regular polygon is $18∘18^\circ$ . Find:

the number of sides
each interior angle

Question 3

Four interior angles of a pentagon are $92∘92^\circ$ , $115∘115^\circ$ , $128∘128^\circ$ and $137∘137^\circ$ . Find the fifth interior angle.

Question 4

The interior angle sum of a polygon is $1260∘1260^\circ$ . Find the number of sides.

Answers to the exam questions

Answer 1

One exterior angle is:

360∘÷12=30∘360^\circ \div 12=30^\circ

Therefore, one interior angle is:

180∘−30∘=150∘180^\circ-30^\circ=150^\circ

Answer: $150∘150^\circ$

Answer 2

The number of sides is:

360∘÷18∘=20360^\circ \div 18^\circ=20

Each interior angle is:

180∘−18∘=162∘180^\circ-18^\circ=162^\circ

Answers: $20$ sides and $162∘162^\circ$

Answer 3

The interior angle sum of a pentagon is:

\times 180^\circ=540^\circ

The known angles total:

92∘+115∘+128∘+137∘=472∘92^\circ+115^\circ+128^\circ+137^\circ=472^\circ

The missing angle is:

540∘−472∘=68∘540^\circ-472^\circ=68^\circ

Answer: $68∘68^\circ$

Answer 4

Form an equation using the interior angle sum:

\times 180=1260

Divide by $180$ :

n - 2 = 7

n = 9

Answer: $9$ sides

Embedded exam-style question

Each interior angle of a regular polygon is $156∘156^\circ$ . Work out the number of sides of the polygon.

First find one exterior angle:

180∘−156∘=24∘180^\circ-156^\circ=24^\circ

Then divide the full turn by the exterior angle:

360∘÷24∘=15360^\circ \div 24^\circ=15

The polygon has $15$ sides.

Try 4 more angles in polygons practice questions, then compare your working with the provided solution and mark scheme. You can also work through the printable angles in polygons exam booklet.

How to revise angles in polygons effectively

Learn the relationships rather than treating each formula as an isolated fact. The triangle angle sum leads to $\times 180^\circ$ , exterior angles make one complete turn of $360∘360^\circ$ , and an interior angle with its exterior angle forms a straight line of $180∘180^\circ$ .

When practising, include questions that ask for different unknowns: a missing angle, one angle of a regular polygon, the number of sides and an algebraic value. Check the wording carefully for regular, because that single word determines whether the angles can be treated as equal.

Angles in polygons appear within the geometry content of GCSE Maths. Use the GCSE Maths topics by exam board and tier guide to check how this topic fits into your wider revision plan.

Conclusion

For angles in polygons GCSE Maths questions, remember three connected facts: an $n$ -sided polygon has an interior angle sum of $\times 180^\circ$ , exterior angles add to $360∘360^\circ$ , and an interior angle plus its corresponding exterior angle equals $180∘180^\circ$ .

Check whether a polygon is regular before dividing an angle sum equally. Show your formula, substitute carefully and confirm that any calculated number of sides is a whole number. The MathsGenie revision lesson, practice questions, exam booklet and mark schemes provide the most useful next steps for consolidating these methods.

Sources and referenced URLs