Angle in a semicircle is 90°

The angle in a semicircle theorem is a circle theorem. This article looks at the theorem itself and how it relates to other circle theorems. It includes a proof of the theorem, and there are several exam-style questions and answers 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 will always be 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:

When the angle is in the major segment, it will be an acute angle (that is, less n 90°). This is shown for angle x of the left of the diagram above.

When the angle is in the minor segment, it will be an obtuse angle (that is, greater that 90°). This is shown for angle y in the centre of the diagram.

When AB is a diameter, the segment is a semicircle, which is neither a major segment nor a minor segment. In that case that angle is always 90°, which is the angle in a semicircle theorem. That is shown for angle z on the right if the diagram.

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 will be 180°. This means that $x$ will be 90°, which is the angle in a semicircle theorem.

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 click the button at the end of each question to reveal the answer. You can also find worked solutions on the GraphicMaths Youtube channel:

The Circle Theorems Workbook contains additional questions and answers related to circle theorems.

Question 1

The line ST 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.

Show answer 1

Question 2

The circle below has a diameter of 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.

Show answer 2

Question 3

The line ST 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.

Show answer 3

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 $5y$, angle UVW is 3y, and angle UWX is x:

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

Show answer 4

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