The right angle in a half circle

The idea

The angle subtended by a diameter at any point on the circle is 90°, since the diameter subtends a straight angle of 180° at the centre.

A set square on a table

A carpenter's set square has a corner of exactly 90°. Imagine the corner moving round the edge of a round table, with its two arms always touching the ends A and B of a diameter.

What is the angle ∠ADB as the point D moves round the circle?

• Always 90°
• It changes as D moves
• It is bigger near A and B

Catch it out

AB is a diameter. Drag D round the circle and read ∠ADB. Try to catch it out.

Half of 180°

The angle subtended by a diameter at any point on the circle is 90°.

A diameter subtends a straight angle, 180°, at the centre. The angle at any point D on the circle is half of that: 180° ÷ 2 = 90°.

Notes

The angle subtended by a diameter at any point on the circle is 90°: it is half of the straight angle, 180°, at the centre.

Check yourself

AB is a diameter of a circle and D is a point on the circle. What is ∠ADB, in degrees?

Answer: 90 °

The angle at the centre is 180°, and the angle at D is half of it: 90°.

AB is a diameter of a circle and D is a point on the circle. ∠DAB = 35°. What is ∠DBA, in degrees?

Answer: 55 °

∠ADB = 90° because AB is a diameter. So ∠DBA = 180° − 90° − 35° = 55°.

AB is a diameter of a circle and D is a point on the circle. What kind of triangle is ABD?

AB is a diameter of a circle, AB = 10 cm, and D is a point on the circle with AD = 6 cm. How long is BD, in cm?

Answer: 8 cm

∠ADB = 90°, so BD² = 10² − 6² = 64 and BD = 8 cm.

• Right-angled at D — correct. Yes. The diameter AB subtends 90° at D.
• Equilateral. Triangle ABD has a 90° angle at D, so its angles cannot all be 60°.
• Obtuse-angled at D. The angle at D is exactly 90°, not more.