How do I find each interior angle of a regular polygon?

Find each exterior angle (360° ÷ number of sides) and subtract it from 180°. For a regular hexagon: 360° ÷ 6 = 60°, so each interior angle is 120°.

Why do a triangle's angles add up to 180°?

If you tear off the three corners and put them side by side, they form a straight line — and angles on a straight line add up to 180°.

Topic guide · JSS 1, JSS 2, JSS 3, SS 2

Geometry

Angles on a line and at a point, angles in triangles and polygons, and circle theorems — explained with diagrams, worked examples and exam-style practice.

Start the Angles in a Triangle lesson →

Geometry explained

Geometry is the maths of shape and space. A handful of angle facts — angles on a straight line, angles in a triangle, angles around a point — unlock most geometry problems in JSS, and the circle theorems do the same in SS 2.

The trick is to spot which fact applies, write an equation, and solve it. MathPlug's triangle explorer lets you drag the corners of a triangle and watch the three angles always add up to 180°.

Key ideas to remember

Angles on a straight line add up to 180°; angles at a point add up to 360°.
Angles in a triangle add up to 180°. An exterior angle equals the sum of the two opposite interior angles.
Sum of interior angles of an n-sided polygon = (n − 2) × 180°; exterior angles always add up to 360°.
The angle at the centre of a circle is twice the angle at the circumference; the angle in a semicircle is 90°.
Opposite angles of a cyclic quadrilateral add up to 180°.

Worked example

Two angles on a straight line are (4x + 5)° and (3x + 49)°. Find x.

Step 1: Form an equation
4x + 5 + 3x + 49 = 180
Step 2: Collect like terms
7x + 54 = 180
7x = 126
Step 3: Divide
x = 18

Final answer: x = 18

Practice questions

Have a go, then open the answer to check your working. Every answer was checked by the MathPlug maths engine.

Question 1
In triangle PQR, ∠P = 75° and ∠Q = 54°. Side QR is extended to S. Find the exterior angle ∠PRS.
- A231°
- B105°
- C129°
- D51°
Show answer and working
1. Step 1: Exterior angle theorem
  ∠PRS = ∠P + ∠Q = 75° + 54° = 129°
2. Step 2: Check
  ∠PRQ = 180° − 75° − 54° = 51°, and 51° + 129° = 180° ✓
Final answer: 129°
Question 2
Each exterior angle of a regular polygon is 45°. How many sides does it have?
- A7
- B4
- C8
- D10
Show answer and working
1. Step 1: Divide 360° by the exterior angle
  360° ÷ 45° = 8 sides
Final answer: 8
Question 3
AB is a diameter of a circle and C is a point on the circumference. If ∠CAB = 32°, find ∠CBA.
- A122°
- B58°
- C148°
- D32°
Show answer and working
1. Step 1: Angle in a semicircle
  ∠ACB = 90°
2. Step 2: Angles in triangle ABC
  ∠CBA = 180° − 90° − 32° = 58°
Final answer: 58°

More practice with instant feedback →

Frequently asked questions

How do I find each interior angle of a regular polygon?: Find each exterior angle (360° ÷ number of sides) and subtract it from 180°. For a regular hexagon: 360° ÷ 6 = 60°, so each interior angle is 120°.
Why do a triangle's angles add up to 180°?: If you tear off the three corners and put them side by side, they form a straight line — and angles on a straight line add up to 180°.