Q1. An arc subtends an angle of 70° at a point on the remaining part of the circle. Find the angle it subtends at the centre.
Angle at the centre = 2 × angle at the circumference
= 2 × 70°
= 140°
Circles — Class 9
A circle is packed with elegant angle relationships. Learn the handful of rules that solve almost every circle question.
Equal chordsEqual chords of a circle are equidistant from the centre, and subtend equal angles at the centre.
Angle subtended by an arcThe angle an arc subtends at the centre is twice the angle it subtends at any point on the remaining part of the circle.
Angle in a semicircleThe angle in a semicircle is always 90°.
Cyclic quadrilateralA quadrilateral inscribed in a circle; its opposite angles are always supplementary (sum to 180°).
Solved examples
Q2. In a cyclic quadrilateral, one angle is 110°. Find the opposite angle.
Opposite angles of a cyclic quadrilateral are supplementary.
180° − 110°
= 70°
Formula sheet
| Quantity | Formula |
|---|---|
| Angle at centre vs circumference | angle at centre = 2 × angle at circumference |
| Angle in a semicircle | = 90° |
| Cyclic quadrilateral | opposite angles sum to 180° |
Common mistakes & exam wins
- The angle in a semicircle is always 90° — a special case of "angle at centre = 2× angle at circumference" (since the centre angle there is 180°).
- Opposite angles of a cyclic quadrilateral sum to 180° — this is often the fastest way to solve circle problems.
- Equal chords are equidistant from the centre, and vice versa.
Frequently asked questions
What is the relationship between the angle at the centre and at the circumference?
The angle subtended by an arc at the centre is twice the angle it subtends at any point on the remaining part of the circle.
What is the angle in a semicircle?
Always 90° (a right angle).
What is a cyclic quadrilateral?
A quadrilateral whose all four vertices lie on a circle; its opposite angles always sum to 180°.