Circle Theorems (Year 11)
Free printable UK Year 11 (GCSE Higher tier) maths worksheet on circle theorems: angle at the centre, angle in a semicircle, same segment, cyclic quadrilaterals and tangent-radius, with accurate diagrams.
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Grade 10 · Math worksheet
Name
Date
Math
Circle Theorems
Each diagram shows a circle with centre O (drawn accurately from the given values, but do not measure from it). Find each marked angle and NAME the circle theorem you used: the reason earns marks at GCSE, not just the number. Higher tier.
- 1.A, B and P lie on a circle with centre O. Angle AOB = 64° (at the centre). Find angle APB, marked x, and give a reason.
- 2.A, B and P lie on a circle with centre O. Angle APB = 39° (at the circumference). Find angle AOB, marked x, and give a reason.
- 3.A, B and P lie on a circle with centre O. Angle AOB = 94° (at the centre). Find angle APB, marked x, and give a reason.
- 4.AB is a diameter of the circle and P lies on the circle. Angle PAB = 40°. Find angle PBA, marked x, and give reasons.
- 5.AB is a diameter of the circle and P lies on the circle. Angle PAB = 55°. Find angle PBA, marked x, and give reasons.
- 6.A, B, P and Q lie on a circle, with P and Q on the same side of the chord AB. Angle APB = 44°. Find angle AQB, marked x, and give a reason.
- 7.A, B, P and Q lie on a circle, with P and Q on the same side of the chord AB. Angle APB = 40°. Find angle AQB, marked x, and give a reason.
- 8.A, B, P and Q lie on a circle, with P and Q on the same side of the chord AB. Angle APB = 47°. Find angle AQB, marked x, and give a reason.
- 9.ABCD is a cyclic quadrilateral. Angle A = 104° and angle B = 92°. Find angle C (marked x) and angle D (marked y), giving a reason.
- 10.ABCD is a cyclic quadrilateral. Angle A = 75° and angle B = 106°. Find angle C (marked x) and angle D (marked y), giving a reason.
- 11.ABCD is a cyclic quadrilateral. Angle A = 114° and angle B = 95°. Find angle C (marked x) and angle D (marked y), giving a reason.
- 12.PT is a tangent to the circle with centre O, touching it at T. Angle OPT = 27°. Find angle TOP, marked x, and give reasons.
- 13.PT is a tangent to the circle with centre O, touching it at T. Angle OPT = 34°. Find angle TOP, marked x, and give reasons.
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