Angles in shapes

Video lesson3 worked examples

Theory

Polygons

The exterior angles of any regular polygon sum to 360°.

Circle Properties

You must use circle properties to find missing angles. Key rules:

A tangent meets a radius at exactly 90°.
An angle in a semicircle (standing on a diameter) is exactly 90°.
The perpendicular bisector of any chord always passes through the centre of the circle.
Any triangle formed by two radii is isosceles, so its base angles are equal.

The Golden Rule: in a circle problem, hunt for the special angles first — a tangent–radius right angle, an angle in a semicircle, or the equal base angles of an isosceles radius triangle. Mark them on the diagram, then chase the missing angle using “angles in a triangle sum to 180°”.

⚠️ Common Examiner Traps

Interior vs exterior: for a regular polygon, find the exterior angle ( $360 \div n$ ) first, then subtract from 180° for the interior.
Two radii are equal: that is why the triangle is isosceles — the two base angles are equal, which is often the key step.
Angle in a semicircle: it is $90^\circ$ only when the triangle stands on a diameter.
Give reasons: circle-geometry questions expect you to name the property you used, not just the number.

Worked examples

Example 1

Regular Polygons

Find the size of one interior angle of a regular decagon (10 sides).

Step 1: Find the exterior angle first: 360° ÷ 10 = 36°.

Step 2: Interior and exterior angles lie on a straight line (180°). Subtract the exterior from 180°: 180° − 36°.

Answer: 144°.

Example 2

Circle Geometry (Tangent)

A tangent touches a circle at point P. The centre of the circle is O and has a radius of 5 cm. A point Q lies on the tangent such that the distance OQ is 13 cm. Find the length of the tangent PQ.

Step 1: Recognise that the radius OP and tangent PQ meet at a right angle (90°), making ΔOPQ a right-angled triangle.

Step 2: Use Pythagoras to find PQ: PQ² = 13² − 5² = 169 − 25 = 144.

Answer: PQ = 12 cm.

Example 3

🎯 Exam-style (angle in a semicircle)

A, B and C lie on a circle, and AB is a diameter. Angle CAB is 34°. Calculate the size of angle ABC.

Step 1: Because AB is a diameter and C is on the circle, angle ACB is an angle in a semicircle, so $\angle ACB = 90^\circ$ .

Step 2: The three angles of triangle ABC add to 180°:

\angle ABC = 180^\circ - 90^\circ - 34^\circ

Answer: angle ABC = 56°.

Ready to practise?

8 Angles in Shapes questions with answers and video solutions