Circle Theorems: Angle in a Semicircle

What’s Special About a Semicircle?

There’s a neat and powerful rule in circle geometry:

The angle in a semicircle is always $90\degree$

This means if you draw a triangle where:

• One side is the diameter of a circle
• The third point lies anywhere on the circumference

Then the angle opposite the diameter is always a right angle.

How It Works

Think of the diameter as creating a semi (half) circle.

If you draw lines from the ends of that diameter to any point on the curve, you’ll always create a right-angled triangle.

In short:

$$\text{Angle in a semicircle} = 90\degree$$

This is actually a special case of another rule:

• The angle at the centre of a circle is double the angle at the edge
• A diameter makes an angle of $180\degree$ at the centre
• So the angle at the circumference is:

$$\frac{180\degree}{2} = 90\degree$$

How to Spot It

Look out for:

• A triangle inside a circle
• One of the sides is a diameter
• All corners (vertices) are on the circle

Then the angle opposite the diameter must be a right angle.

Example

Question:

Points P,Q,RP, Q, R lie on a circle.

Line RQRQ is the diameter, and the triangle PQRPQR is drawn inside the circle.
You are told:

$$\angle PQR = 40\degree, \quad \angle PRQ = y\degree$$

Find the value of $y$, and give a reason for your answer.

Step 1: Use the circle theorem:

$$\angle PRQ = 90\degree \quad \text{(angle in a semicircle)}$$

So we now know:

$$\angle PRQ = 90\degree, \quad \angle PQR = 40\degree$$

Step 2: Use the triangle angle rule:

Angles in a triangle add up to $180\degree$

Write an equation:

$$y + 90 + 40 = 180$$

Solve:

$$y = 180 - 130 = 50\degree$$

Final Answer:

$$\boxed{y = 50^\circ}$$

Reason:

The angle in a semicircle is $90\degree$
 

Worked Example

In a circle with diameter $AB$, point $C$ lies on the circumference forming triangle $ABC$.

If $\angle ABC = 65\degree$, what is $\angle CAB$ ?

Give your answer and explain your reasoning using a circle theorem.

Solution:

Since $AB$ is the diameter, the angle opposite it ($\angle ACB$) is $90\degree$.

Use triangle angle facts:

$$\angle CAB = 180\degree - 90\degree - 65\degree = 25\degree$$

$$\boxed{\angle CAB = 25\degree}$$

Reason:

The angle in a semicircle is $90\degree$; angles in a triangle sum to $180\degree$