What is the sum of interior angles in a triangle?
Basic rule.
Angles in a triangle add to 180°.
Polygons
Polygons are closed 2D shapes made from straight sides. Their angle rules are essential for GCSE Maths and link closely to angles, algebra and forming equations
Overview
A polygon is a 2D shape made only from straight sides.
Triangles, quadrilaterals, pentagons and hexagons are all polygons.
Sum of interior angles of an \(n\)-sided polygon: \( (n - 2)\times180^\circ \)
In GCSE Maths, polygon questions often ask you to identify the shape, find interior angle sums, work with regular polygons, or calculate exterior angles.
What you should understand after this topic
- Understand what a polygon is
- Know the names of common polygons
- Understand the difference between regular and irregular polygons
- Find interior angle sums
- Work with exterior angles of regular polygons
Key Definitions
Polygon
A 2D shape made only from straight line segments.
Regular Polygon
A polygon with all sides equal and all angles equal.
Irregular Polygon
A polygon where sides or angles are not all equal.
Interior Angle
An angle inside the polygon.
Exterior Angle
An angle outside the polygon, formed by extending a side.
Side
One straight edge of the polygon.
Key Rules
Triangle
3 sides
Quadrilateral
4 sides
Pentagon
5 sides
Hexagon
6 sides
Heptagon
7 sides
Octagon
8 sides
Nonagon
9 sides
Decagon
10 sides
Important Formulas
Interior angle sum
\( (n - 2) \times 180^\circ \)
Exterior angle sum
\( 360^\circ \)
One exterior angle of a regular polygon
\( \frac{360^\circ}{n} \)
One interior angle of a regular polygon
\( 180^\circ - \text{exterior angle} \)
How to Solve
Step 1: Recognise a polygon
A polygon is a closed 2D shape made only from straight sides.
Triangles, quadrilaterals, pentagons and hexagons are polygons.
Circles and shapes with curved edges are not polygons.
Exam tip: Curved sides mean the shape is not a polygon.
Step 2: Know common polygon names
3 sides
Triangle
4 sides
Quadrilateral
5 sides
Pentagon
6 sides
Hexagon
7 sides
Heptagon
8 sides
Octagon
Step 3: Regular or irregular
A regular polygon has equal sides and equal angles.
Regular shapes often appear with angle rules in exam questions.
Regular polygon
All sides equal and all angles equal.
Irregular polygon
At least one side or angle is different.
Step 4: Find the interior angle sum
The total of the interior angles depends on the number of sides.
\( \text{Interior angle sum} = (n - 2) \times 180^\circ \)
\(n\) is the number of sides.
Exam tip: Use this for any polygon, regular or irregular.
Step 5: Use exterior angles
The exterior angles of any polygon always add to \(360^\circ\).
\( \text{Sum of exterior angles} = 360^\circ \)
Key idea: This works for all polygons.
Step 6: Regular polygon angles
For regular polygons, all exterior angles are equal.
\( \text{One exterior angle} = \frac{360^\circ}{n} \)
\( \text{One interior angle} = 180^\circ - \text{exterior angle} \)
Exam thinking: Find the exterior angle first, then the interior angle.
Step 7: Find the number of sides
You can work backwards if one exterior angle is given.
\( n = \frac{360^\circ}{\text{one exterior angle}} \)
This only works directly for regular polygons.
Step 8: Exam method summary
- Count the number of sides.
- Decide whether the polygon is regular or irregular.
- Use \((n - 2) \times 180^\circ\) for total interior angle sum.
- Use \(360^\circ \div n\) for one exterior angle of a regular polygon.
- Use \(180^\circ - \text{exterior angle}\) for one interior angle.
Example Questions
Edexcel
Exam-style questions focusing on naming polygons and recognising polygon angle facts.
Edexcel
The polygon shown has 6 sides.
Name the polygon.
Edexcel
The exterior angles of a polygon are shown.
What is the sum of the exterior angles of any polygon?
AQA
Exam-style questions focusing on interior and exterior angles of regular polygons.
AQA
A pentagon is shown.
Find the sum of the interior angles.
AQA
A regular decagon has 10 equal exterior angles.
Find one exterior angle.
OCR
Exam-style questions focusing on higher polygon angle reasoning.
OCR
A regular nonagon has 9 equal exterior angles.
Find one interior angle of the regular nonagon.
OCR
A regular polygon has exterior angle 24°.
Find the number of sides.
OCR
A regular polygon has interior angle 150°.
Find the exterior angle, then find the number of sides.
Exam Checklist
Step 1
Check how many sides the polygon has.
Step 2
Decide whether the shape is regular or irregular.
Step 3
Choose the correct formula: interior sum, exterior sum, or one angle.
Step 4
Check whether the answer is sensible for the shape.
Most common exam mistakes
Wrong formula
Using the interior sum formula when the question asks for one angle.
Wrong subtraction
Forgetting that one interior angle comes from \(180^\circ - \text{exterior angle}\).
Regular polygon mistake
Using equal-angle methods on irregular polygons.
Name confusion
Mixing up pentagon, hexagon, heptagon and octagon.
Common Mistakes
These are common mistakes students make when working with polygons in GCSE Maths.
Using the wrong value for n
Incorrect
A student uses an incorrect number of sides in formulas.
Correct
In polygon formulas, \(n\) represents the number of sides. Make sure you count the sides correctly before substituting.
Confusing total interior angle sum with one angle
Incorrect
A student calculates the total interior angle sum and treats it as a single angle.
Correct
The formula \((n - 2) \times 180\) gives the total interior angle sum. To find one interior angle in a regular polygon, divide by \(n\).
Forgetting exterior angles sum to 360°
Incorrect
A student does not use the exterior angle rule.
Correct
The exterior angles of any polygon always add up to \(360^\circ\), which can be used to find one exterior angle in a regular polygon.
Using regular polygon rules on irregular shapes
Incorrect
A student applies formulas assuming all sides and angles are equal.
Correct
Formulas for finding one interior or exterior angle only apply to regular polygons. Irregular polygons require different methods.
Using 360° instead of 180° incorrectly
Incorrect
A student subtracts from \(360^\circ\) when finding an interior angle.
Correct
Interior angles relate to \(180^\circ\) at a point on a straight line. For a regular polygon, one interior angle = \(180^\circ - \text{exterior angle}\).
Try It Yourself
Practise calculating interior and exterior angles of polygons.
Foundation
Foundation Practice
Use angle facts for triangles, quadrilaterals and simple polygons.
Find the sum of interior angles of a quadrilateral.
Think two triangles.
A quadrilateral can be split into 2 triangles → 2 × 180 = 360°.
Find the size of each interior angle in a regular triangle.
Divide total by number of sides.
180 ÷ 3 = 60°.
Find the sum of interior angles of a pentagon.
Use (n − 2) × 180.
(5 − 2) × 180 = 540°.
Find each interior angle of a regular square.
Square has 4 equal angles.
360 ÷ 4 = 90°.
Find the sum of interior angles of a hexagon.
Use (n − 2) × 180.
(6 − 2) × 180 = 720°.
The sum of exterior angles of any polygon is:
Full turn.
Exterior angles always add to 360°.
A regular pentagon has equal angles. Find each interior angle.
540 ÷ 5.
Each angle = 540 ÷ 5 = 108°.
Which shape has the largest interior angle?
More sides = larger angles.
More sides increases interior angle size.
Find the sum of interior angles of a heptagon.
(n − 2) × 180.
(7 − 2) × 180 = 900°.
Higher
Higher Practice
Solve algebraic and multi-step polygon angle problems.
Find each interior angle of a regular octagon.
Total ÷ number of sides.
1080 ÷ 8 = 135°.
Find the number of sides of a polygon with interior angle 120°.
Use interior angle formula.
120° corresponds to a hexagon.
Find the number of sides of a polygon with each exterior angle 40°.
360 ÷ exterior angle.
360 ÷ 40 = 9 sides.
A polygon has interior angle 150°. How many sides does it have?
Use formula or reverse process.
150° gives 12 sides.
Which statement is true?
Think full turn.
Exterior angles sum to 360°.
The interior angles of a polygon add to 1440°. How many sides does it have?
(n − 2) × 180 = 1440.
1440 ÷ 180 = 8 → n = 10.
Find the exterior angle of a regular hexagon.
360 ÷ 6.
Each exterior angle = 60°.
A regular polygon has exterior angle 24°. Find the number of sides.
360 ÷ 24.
360 ÷ 24 = 15 sides.
A polygon has 20 sides. Find the sum of interior angles.
(n − 2) × 180.
(20 − 2) × 180 = 3240°.
Each interior angle of a regular polygon is 165°. Find the number of sides.
Exterior angle = 180 − 165.
Exterior = 15°, so 360 ÷ 15 = 24 sides.
Games
Practise this topic with interactive games.
Games coming soon.
Frequently Asked Questions
How do I find interior angles?
Use the formula (n−2) × 180.
What is an exterior angle?
The angle outside a polygon.
What do exterior angles add up to?
360 degrees.