Home → Curriculum Hub → Maths Lessons → Schemes of Work → Three-Year GCSE Higher → Circle Theorems
This Higher GCSE scheme of work unit covers circle theorems in Year 10. It collects the circumference theorems, then combines two of them on one diagram, then meets the tangent facts, and finishes by chaining a tangent theorem with an earlier circumference fact.
Students have to write the theorem name beside each new angle, and they have to mark equal radii before any isosceles triangle is used. Those two habits are what later combined Higher circle diagrams all use.
A student who has secured this unit:
Students should be secure with the following before beginning this unit.
The angle at the centre is twice the angle at the circumference standing on the same arc. Swapping the two angles doubles or halves the wrong way.
An angle in a semicircle, standing on a diameter, is 90°. The diameter is the hypotenuse of that triangle.
Angles in the same segment are equal. Opposite angles of a cyclic quadrilateral sum to 180°.
A tangent is perpendicular to the radius at the point of contact. Two tangents from an external point are equal. The alternate-segment theorem equates the angle between tangent and chord with the angle in the opposite segment.
| Misconception | Teaching focus |
|---|---|
| Treating the sketch as accurate and measuring an angle instead of naming a theorem. | Cover the picture and ask which named theorem gives the angle without a protractor. |
| Using the centre-circumference theorem on two angles that do not stand on the same arc. | Trace the shared arc in colour and refuse the doubling until both angles sit on that arc. |
| Calling a chord a diameter, so an acute angle on the circumference is forced to 90°. | Ask whether the chord passes through the marked centre before the semicircle theorem is used. |
| Forgetting that opposite angles in a cyclic quadrilateral sum to 180°, and using co-interior language instead. | Tick the four vertices on the circumference, then write the pair that sits opposite. |
| Applying the alternate-segment theorem to the angle in the same segment as the tangent-chord angle. | Shade the opposite segment first, and only then copy the tangent-chord angle into that region. |
Every lesson in this unit is already built. A Mr Mathematics membership gives you the presentation, worksheet, question generator and video tutorial for all 4 lessons, alongside the full Key Stage 3, GCSE, IGCSE and A-Level libraries.
A school membership covers every teacher in your department, so the whole scheme of work is resourced from one subscription.
Students are taken through the discovery of various circle theorems.
The theorems include, angle at the centre is twice the angle at the circumference, angles in the same segment and angles in cyclic quadrilaterals.
Students learn to apply multiple circle theorems to solve complex problems.
Later, students combine the multiple circle theorems to derive proof as learning progresses.
Students discover the alternate segment theorem and angle between the radius and tangent of a circle.
Later, as learning progresses, they apply these theorems to work out unknown angles.
Students learn how to solve problems involving the angle between radius and tangent and the alternative segment theorem.
Later, they combine multiple angle properties as learning progresses to prove the alternate segment theorem.
Applying Circle Theorems - Part 1
Applying Circle Theorems - Part 2
Applying Circle Theorems - Part 3
Students gain additional practice applying multiple circle theorems.
The problem-solving activity challenges students to apply Pythagoras' Theorem and derive a proof.
Students solve problems with circle theorems by linking them to trigonometry, Pythagoras' Theorem, 2D shape areas, and perimeters.
As learning progresses, they apply the intersecting chords property and similar shapes to set up and solve equations, deepening their understanding of geometric relationships.
Applying Circle Theorems - Part 1
Applying Circle Theorems - Part 2
Applying Circle Theorems - Part 3
Students revise how to find missing angles in circles and tangents.
Theorems include angles at the centre and circumference, angles in the same segment, cyclic quadrilaterals, angles between a tangent and chord and the alternate segment theorem.
Students learn how to apply circle theorems to find unknown angles in diagrams. They begin by identifying key facts such as angles in the same segment, angles at the centre, and cyclic quadrilaterals.
As learning progresses, they justify angle relationships with clear reasoning, combine multiple theorems in a single diagram, and solve multistep problems involving tangents and radii.
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