Angle in a semicircle is 90°

The angle in semicircle theorem is a circle theorem. This article looks at the theorem itself, it's applications and proof, and there are several exam-style questions at the end.

We can draw a triangle ABC where AB is a diameter of a circle, and C lies on the circumference of the circle:

The angle at the circumference C is a right angle. We say the angle in a semicircle is a right angle.

Here is a video on this topic:

Relationship to other circle theorems

The angle in same segment theorem says that for any chord AB, every angle in the same segment of that chord is equal:

In the case where AB is a diameter, that angle is always 90°.

The angle at centre theorem says that the angle formed by a chord at the centre of a circle is twice the angle at the circumference. In the diagram below, the angle at the centre is 2x and the angle at the circumference is x:

Notice that when AB is a diameter, the angle AOB is 180°, so $x$ is 90°.

Proof

We can prove this as follows.

We start by drawing an extra line from the centre O to the point C.

Notice that the lines OA, OB and OC are all radii of the circle, and therefore all of equal length.

Looking at the triangle AOC, this is an isosceles triangle (from the rule 2 radii form an isosceles triangle). So the two angles at the circumference are equal (we will call them a):

By the same logic, the two angles at the circumference in triangle BOC are equal, and we will call them b:

Looking at the original triangle ABC:

The angle at A is a, the angle at B is b, and the angle at C is a+b. Since the three angles of a triangle add up to 180° we have:

Gathering the terms a and b:

Dividing both sides by 2 gives:

Since the angle at C is a + b, this proves that the angle at C is 90°.

Questions

Here are some exam-style questions on the angle in semicircle theorem. You can find worked solutions on the GraphiMaths Youtube channel:

Question 1 The line TS is a diameter of a circle, centre O. Point R is on the circumference of the circle. Angle RTS is 34°:

What is angle $x$? Show your working.

Question 2 The circle below had diameter 5 and centre O. The line BC is a diameter of the circle,. Point A is on the circumference of the circle. The side AB has length 3:

What then length of side AC, marked as $x$? Show your working.

Question 3 The line TS is a diameter of a circle, centre O. Point R is on the circumference of the circle. Angle RTS is $x$ and angle RST is 5x:

What is $x$? Show your working.

Question 4 The line VW is a diameter of a circle, centre O. Point U is on the circumference of the circle. Angle WUV is $4y$, angle UVW is 3y, and angle UWX is x:

What are $x$ and $y$? Show your working.

See also

• Parts of a circle
• Perpendicular bisector of chord theorem
• Angle at the centre of a circle is twice the angle at the circumference
• Angles in the same segment of a circle are equal
• Opposite angles in a cyclic quadrilateral add up to 180°
• Tangent and radius of a circle meet at 90°
• Two tangents from a point have equal length
• Alternate segment theorem
• Two radii form an isosceles triangle