> ## Content Index > Fetch the complete content index at: https://www.esheets.io/llms.txt > Use this file to discover other available public pages before exploring further. # Circle theorems - URL: https://www.esheets.io/circle-theorems/ - Published: 2026-10-04T10:37:07.000Z - Updated: 2026-10-04T10:37:07.000Z - Author: Richard Linnington - Tags: Geometry and Shape, Maths Practise circle theorems for GCSE Higher with 18 questions on angles, tangents and chords. Start with single-theorem questions, then combine circle facts, algebra and triangle rules in exam-style problems. There is also a little length work, including Pythagoras. [Jump to the questions](#practise-now) ## Practise now Worksheet preview and key skills ### Worksheet preview Work through 18 GCSE Higher questions, worth 18 marks altogether. Each question has a circle diagram and one numeric answer: an angle, a length or a value of x. The first eight questions cover centre and circumference angles, angles in the same segment, semicircles and cyclic quadrilaterals, including simple algebra. Questions 9–14 introduce tangents, the alternate segment theorem and chord lengths. The final four combine theorems in multi-step exam-style problems. #### What you’ll practise - Choosing when to double, halve, equate or subtract angles. - Using radius–tangent perpendicularity, equal tangents and the alternate segment theorem. - Bisecting a chord with a perpendicular from the centre and using Pythagoras to find its full length. - Combining cyclic quadrilaterals, central angles and isosceles triangles. Select Check answer for numeric feedback, or use New Questions for a fresh set. Teacher reveal includes the answers and reasons; written reasons are not automatically marked. Use the interactive worksheet below, or read the [Topic guide](#topic-guide) for the method and worked examples. ## Circle theorems Where shown, the dashed arc marks the relevant angle. Find the requested angle, length or value of x. Enter a number only, then select Check answer. Each question is worth 1 mark; use the stated facts rather than measuring diagrams. ## Topic guide Circle theorems connect angles and lengths in circles. A chord joins two points on a circle; a diameter is a chord through the centre. A tangent touches the circle at one point. A cyclic quadrilateral has all four vertices on the circle. ### Eight circle facts to recognise 1. **Centre and circumference:** the angle at the centre is twice the angle at the circumference standing on the same arc. Check the endpoints and which arc is intercepted before doubling or halving. 2. **Same segment:** angles at the circumference standing on the same chord, with their vertices in the same segment, are equal. 3. **Semicircle:** the angle at the circumference standing on a diameter is 90°. 4. **Cyclic quadrilateral:** opposite angles sum to 180°. 5. **Radius and tangent:** a radius meets a tangent at 90° at the point of contact. 6. **Equal tangents:** the two tangent segments from the same external point to their points of contact have equal lengths. 7. **Centre to chord:** a perpendicular from the centre to a chord bisects the chord. Both the centre and the right angle matter. 8. **Alternate segment:** the angle between a tangent and a chord equals the angle at the circumference in the alternate segment, standing on that chord. **Supporting facts:** radii of the same circle are equal, so a triangle formed by two radii is isosceles and its two base angles are equal. Angles in a triangle sum to 180°, and angles on a straight line sum to 180°. A right-angled triangle may also allow you to use Pythagoras. ### Choosing a route Identify the required angle or length first. Look for a centre, diameter, tangent, cyclic quadrilateral or marked perpendicular. Write down a value you can deduce, then look again: that new value often unlocks the next theorem. In an algebra question, form an equation using the angle relationship before solving for x. Give the requested value, which may be x rather than the size of an angle. ### Worked angle example O is the centre and angle AOB is 126°. C lies on the major arc AB, so angle ACB stands on the same minor arc AB as angle AOB. The angle at the circumference is half the centre angle: **angle ACB = 126 ÷ 2 = 63°**. ### Worked tangent example A tangent at A makes an angle of 47° with chord AB. C is on the circumference in the alternate segment. The alternate segment theorem gives angle ACB = 47°. If angle BAC = 61°, the triangle sum gives **angle ABC = 180 − 47 − 61 = 72°**. Use the tangent angle belonging to chord AB, not a different chord through A. ### Worked chord and Pythagoras example O is the centre, OA = 15 cm, and OM = 9 cm is perpendicular to chord AB. The perpendicular bisects AB, so AM = MB. In right-angled triangle OMA, AM² = OA² − OM² = 15² − 9² = 144\. Hence AM = 12 cm and **AB = 2 × 12 = 24 cm**. Pythagoras gives the half-chord first. ### Worked mixed example PA and PB are tangents from P, angle APB = 74°, and O is the centre. Find angle ACB, where C lies on the major arc AB. 1. The tangent contacts give two right angles: OAP = OBP = 90°. 2. The angles in quadrilateral OAPB sum to 360°, so angle AOB = 360 − 90 − 90 − 74 = 106°. 3. Now the centre angle is known. On the same arc, the circumference angle is half as large: **angle ACB = 53°**. ### Common mistakes - Doubling when you should halve: decide whether you are finding the centre or circumference angle. - Using same-segment equality for vertices on opposite sides of the chord. - Assuming a right angle without a diameter for the semicircle theorem. - Treating any quadrilateral as cyclic: all four vertices must lie on the circle. - Forgetting the right angle between a radius and tangent at their contact point. - Confusing equal tangents with equal radii: identify the actual segments. - Using the alternate segment theorem on the wrong chord or with the wrong tangent ray. - Bisecting a chord without checking that the perpendicular comes from the centre. - Matching a centre angle to a circumference angle on the wrong arc, or confusing the minor and reflex centre angles. - Stopping after the first theorem when the question needs a second or third step. The worksheet marks only the requested numeric answers. Practise naming the theorem at each step on paper; teacher reveal supplies concise working. Proving the circle theorems is a separate skill and is not the focus of this worksheet.