Angle at the centre
A(5, 0), B(0, 5): angle AOB = 90°
P on the major arc: angle APB = 90 ÷ 2 = 45°
Reason: angle at the centre is twice the angle at the circumference
Circle theorems What stays the same as a point moves round a circle?
Higher · AQA · 45 minutes
Students move a point round a circle and see an angle that never changes, then meet the eight theorems on a grid where every chord ends at a whole-number point. Each comes with the exact reason to write. Slides, a vote that shows who's stuck, practice and homework.
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A(−5, 0) and B(5, 0) are the ends of a diameter of the circle x² + y² = 25 (top half drawn). Move P round it: what happens to angle APB? Put P on a marked whole-number point and check by counting squares. Then look at the radius OP: what kind of triangles does it cut APB into?
Whole-number points: (−5, 0), (−4, 3), (−3, 4), (0, 5), (3, 4), (4, 3), (5, 0)
The idea in one minute
The angle at the centre is twice the angle at the circumference standing on the same arc. From it follow angles in the same segment (equal), the angle in a semicircle (90°) and the opposite angles of a cyclic quadrilateral (adding up to 180°). A tangent is perpendicular to the radius at the point it touches, two tangents from one point are equal in length, the perpendicular from the centre to a chord cuts it in half, and the angle between a tangent and a chord equals the angle in the alternate segment.
Every diagram is drawn on the circle x² + y² = 25, whose whole-number points ((3, 4), (−4, 3), (0, −5) and the rest) lie exactly on it, so a right angle or two equal tangents can be checked by counting squares, not measured. Marks in circle questions are mostly lost on reasons, so each theorem comes with the sentence to write, and students sort situations into the theorem that answers them before they calculate.
By the end, students can
- use the eight circle theorems to find angles and lengths
- pick the theorem that fits a diagram, and name it in the words a mark scheme wants
- use radii to spot isosceles triangles, and chain two or three reasons
- explain why the angle in a semicircle is 90°
Key words
- chord, arc, segment
- tangent, radius
- circumference
- cyclic quadrilateral
- subtend (stand on)
Before this lesson
Students should know the angle facts (on a line, in a triangle, in a quadrilateral, isosceles triangles) and the words radius, diameter and chord.
Taught in Angles, parallel lines and polygons.
The lesson, step by step
30 minutes in total. Each step says what changes for this length.
3 min 1 · Hook · students think
Move P: what stays the same?
Students check an angle in a semicircle at two points, before anyone explains.
For this length: Check one point only, then move p
Say“A and B are the ends of a diameter. P is on the circle. Work out angle APB at two places without a protractor: count squares.”
Ask- How can you tell an angle is 90° from the grid?
- Where could P go to make the angle bigger?
Watch forMeasuring or guessing the angle: ask how the grid shows a right angle (a quarter turn swaps the two steps of a line and changes one sign).
For the teacher
Show the slide: the circle x² + y² = 25, A(−5, 0), B(5, 0) and P(3, 4). From P, B is 2 right and 4 down. A quarter turn of that step is 4 left and 2 down, and A is exactly twice that (8 left, 4 down). So PA is a quarter turn from PB: angle APB = 90°.
Move p on the slide (or the curve on the page). The angle stays 90° wherever P is. Ask for a reason: draw the radius OP and look at the two triangles it makes.
Gather predictions with Ask the class before moving p on the slide. Keep the word "always": the next step says why.
Shorter: Check one point only, then move p (−2 min).
11 min 2 · Explain · you explain
Eight theorems, eight reasons
The words, then each theorem on the grid with the exact reason to write.
For this length: Show the words slide briefly; do the tangents and the alternate segment on one slide
Say“Each theorem is a picture to spot and a sentence to write. The sentence earns the mark.”
Ask- Which arc does this angle stand on?
- Is that corner on the circle, or is it the centre?
- Where are the radii? What do two radii make?
Watch forCalling the alternate segment theorem "alternate angles": that fact is about parallel lines.
A cyclic quadrilateral
Opposite angles 135° and x
x = 180 − 135 = 45°
Reason: opposite angles of a cyclic quadrilateral add up to 180°
For the teacher
First slide: the words. Chord, tangent, arc, segment, cyclic quadrilateral; an angle named with three letters, the middle one its corner.
Second slide: A(5, 0) and B(0, 5). The radii OA and OB make 90° (along the axes). P(−3, −4) and Q(−4, 3) both give 45°: the angle at the centre is twice the angle at the circumference, and angles in the same segment are equal. With a diameter, the angle at the centre is 180°, so the angle in a semicircle is 90°.
Third slide: four corners on the circle, angles 135°, 135°, 45°, 45°. Opposite angles add up to 180°.
Fourth slide: the tangent at (5, 0) is x = 5, at (0, 5) is y = 5. Each is perpendicular to its radius, and they meet at T(5, 5): TA = TB = 5, two tangents from a point are equal.
Fifth slide: the chord from A(5, 0) to B(−3, 4), and the line from O to M(1, 2). A to M is 4 left and 2 up, and so is M to B: M is the midpoint. A quarter turn of O to M (1 right, 2 up) is 2 left, 1 up, the chord's own direction: OM is perpendicular to the chord. The perpendicular from the centre to a chord bisects the chord. With a radius it makes a right-angled triangle, which is how chord lengths are found (Pythagoras).
Sixth slide: the tangent y = −5 at A(0, −5) and the chord to B(5, 0) make 45°; C(−3, 4), in the other segment, gives angle ACB = 45° too: the alternate segment theorem.
Last two slides: the eight reasons word for word, and the isosceles triangle from two radii, which most longer questions need.
On each slide, ask which arc or chord the angles stand on before saying the theorem. Leave the reasons slides up while they practise.
Shorter: Show the words slide briefly; do the tangents and the alternate segment on one slide (−4 min).
5 min 3 · Your turn · students write
Worked, then yours
Each worked example beside a twin that students solve themselves, reason included.
For this length: Do the first two pairs only
Say“Watch me do one, saying the reason first. Then you do its twin.”
Ask- Which theorem? Say it before you write a number.
Watch forDoubling when the angle at the centre is given: the circumference angle is half of it.
Centre and circumference
Angle ACB = 38°, on arc AB. Find angle AOB.
AOB = 2 × 38 = 76°
Angle at the centre is twice the angle at the circumference
Then theirs
Angle AOB = 124°, on arc AB. Find angle ACB.
Cyclic quadrilateral
Cyclic ABCD: A = 2x + 10, C = x + 50. Find x.
3x + 60 = 180, so x = 40
Opposite angles of a cyclic quadrilateral add up to 180°
Then theirs
Cyclic ABCD: B = 3x − 20, D = x + 60. Find x.
Two tangents
Tangents TA and TB; angle ATB = 50°. Find angle AOB.
OAT = OBT = 90° (tangent perpendicular to radius)
AOB = 360 − 90 − 90 − 50 = 130° (angles in a quadrilateral)
Then theirs
Tangents TA and TB; angle ATB = 64°. Find angle AOB.
For the teacher
Work each example aloud, the reason said before the number. Students solve its twin while you look at who writes the reason.
The tangents pair needs two reasons: the tangent is perpendicular to the radius, and the angles of a quadrilateral add up to 360°.
Open the answers slide once most have finished their twins. A twin with no reason is not finished.
Shorter: Do the first two pairs only (−3 min).
6 min 4 · Who's stuck? · students vote
Vote, then practise
One question shows who's stuck; then practice in levels, and a stretch.
For this length: After the vote, the first six questions only
Say“Choose an answer on your own. Nobody talks until everyone has chosen.”
Ask- Which angle is at the centre, and which is on the circle?
Watch forChoosing a theorem from how the diagram looks, not from where the corners are.
Give each angle as a number of degrees. Write the reason for each step on paper. The stretch is optional.
The question that shows who's stuck
- O is the centre of a circle. A, B and P are on the circle, with P on the major arc AB. Angle AOB = 110°. What is angle APB?
In your Classroom, as they answer (an example) - A4
- B6
- C6
- D12
Each wrong answer is a different mistake. While they answer, your Classroom shows who chose which, so you know in seconds whom to help with what.
- AB is a diameter of a circle, and C is another point on the circle. Angle CAB = 34°. Find angle ABC.
- Which theorem answers each situation? O is the centre of the circle each time. Put each one in the group of the theorem you would use.They sort 10 items into Angle at the centre is twice the angle at the circumference and Angles in the same segment are equal and The angle in a semicircle is 90° and Opposite angles of a cyclic quadrilateral add up to 180° and A tangent is perpendicular to the radius.
- The tangent at A and the chord AB make an angle of 70°. C is on the circle, in the other segment, and angle ACB = 70°. Which reason earns the mark?
- A and B are on a circle with centre O. Why is angle OAB equal to angle OBA?
- The circle x² + y² = 25 has its centre at (0, 0). The tangents at (−5, 0) and at (0, 5) meet at a point T. Move the point to T.They move a point on a grid.
- TA and TB are tangents from a point T, touching the circle at A and B. Angle ATB = 48°. Find angle TAB.
- ABCD is a cyclic quadrilateral. Angle A = 3x + 15 and angle C = 2x − 10, in degrees. They are opposite angles, so they add up to 180°. Solve the equation line by line to find x. Write the numbers without the ° sign.They solve 3x + 15 + 2x − 10 = 180 line by line, each line checked as they write it.
- O is the centre. A, B and C are on the circle, with C on the major arc AB. Angle AOB = 3x + 10 and angle ACB = x + 20, in degrees. Find x.
- The tangent at A makes an angle of 64° with the chord AB. C is on the circle, in the other segment. Angle ABC = 52°. Find angle BAC.
- A circle with centre O has radius 13 cm. The chord AB is 24 cm long. M is the point where the perpendicular from O meets AB. Find the distance OM, in cm.
- Stretch: the tangent to x² + y² = 25 at (3, 4) is perpendicular to the radius from (0, 0) to (3, 4). Move the point to where the tangent crosses the line x = 7.They move a point on a grid.
Support · For the sorting question: for each situation, find the corners first. Is one at the centre? Are four on the circle? Is there a diameter, or a line touching the circle?
For the teacher
Everyone chooses before anyone explains. Then ask a student for each answer to say which theorem they used and where its corners are.
The first questions practise one theorem each and the sort practises choosing one; then the reasons, the tangents on a grid, an equation solved line by line, chains of two reasons and a chord found with Pythagoras. The stretch puts a tangent on the grid by gradients.
Students write their reasons on paper or the board; the quiz keeps their final answers.
If the vote splits, ask two students who chose differently to explain, then sketch the angles on one arc together before anyone practises.
Shorter: After the vote, the first six questions only (−6 min).
Longer: The stretch, then: prove the angle at the centre is twice the angle at the circumference, with two isosceles triangles (+10 min).
5 min 5 · Exit · students show
A chain of reasons
One angle found through two facts, each reason written, shows who can do both.
Say“Find the angle, and write a reason for every step as a full sentence.”
Watch forThe right number with the sum as the only reason, and no theorem in words.
Find the angle and write a reason for each step, as a full sentence.
- TA is a tangent to a circle with centre O, touching it at A. Angle ATO = 28°. Find angle AOT, and write a reason for each step.
For the teacher
Students answer the exit question on their own: one angle, two reasons.
Then two or three read their reasons aloud; the class says whether each would earn the mark.
Read the reasons: start the next lesson with two that would lose the mark, and ask the class to fix them. Set the homework.
Longer: Why is the angle in a semicircle 90°? Explain with the radius to P, as in the hook (+5 min).
One to one: before any number, ask the student to point to the arc or chord an angle stands on and say which theorem fits. Ask them to mark every radius in the diagram first: each pair of radii is an isosceles triangle.
Homework: angles and reasons
Find each angle or letter, and write a reason for every step as a full sentence (for example, "angles in the same segment are equal"). O is always the centre.
- A, B, C and D are on a circle, with C and D on the same side of chord AB. Angle ACB = 41°. Find angle ADB and angle AOB.
- AB is a diameter, and C is on the circle. Angle CAB = 2x and angle CBA = 3x. Find x.
- ABCD is a cyclic quadrilateral. Angle B = 4x + 20 and angle D = x + 10. Find x.
- TA and TB are tangents from T. Angle ATB = 2x and angle AOB = 3x + 40. Find x.
- The tangent at A makes an angle of 3x with the chord AB. C is in the other segment, and angle ACB = x + 36. Find x and the angle.
- Prove that the angle in a semicircle is 90°. (Draw the radius to the third point, and look for isosceles triangles.)
Print the worksheet with working space and homework. Teacher answers come with your Library copy.
Teaching circle theorems: common questions
What reasons do examiners accept for circle theorems?
The theorem in words: "the angle at the centre is twice the angle at the circumference", "angles in the same segment are equal", "the angle in a semicircle is 90°", "opposite angles of a cyclic quadrilateral add up to 180°", "a tangent is perpendicular to the radius", "tangents from a point are equal in length", "the perpendicular from the centre to a chord bisects the chord", "the alternate segment theorem". Shorthand such as "bow tie" or "arrowhead", or the sum on its own, does not earn the reason mark.
How do students choose which circle theorem to use?
Look at the corners, not the overall picture. A corner at the centre: angle at the centre. Two angles on the circle standing on one chord: same segment. A diameter: semicircle. Four corners on the circle: cyclic quadrilateral. A line touching the circle: tangent and radius, two tangents, or the alternate segment. A line from the centre to a chord, or a chord's length: the perpendicular bisects it, then Pythagoras. The lesson's sorting question practises exactly this before any calculating.
Why draw circle theorems on a coordinate grid?
The circle x² + y² = 25 passes exactly through twelve whole-number points, such as (3, 4) and (−4, 3). Chords between them let students check a right angle or two equal tangents by counting squares instead of measuring, so they see that the theorem is true before they use it. It also links to circle equations and perpendicular gradients for Higher students.
Which circle theorems are on GCSE Foundation?
None to prove or apply as theorems: circle theorems are Higher tier. Foundation students need the words (radius, chord, tangent, arc, sector, segment). This lesson is written for Higher.
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Yes. Read the lesson, print the worksheet and open its slides without signing in; nothing you draw there is kept. A free teacher account keeps the whole lesson in your Library to change and teach, with the quizzes, the answers and the answer slides.
Curriculum reference: AQA GCSE Mathematics 8300: G9 (circle words: centre, radius, chord, diameter, circumference, tangent, arc, sector, segment) and G10 (Higher: apply and prove the standard circle theorems on angles, radii, tangents and chords).
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