Circle theorems you can drag: watch every angle rule stay true

How to use this simulation

1. Pick one of the seven theorems under “Theorem”; the legend line below the diagram prints its statement in words.
2. Drag the red points around the circle with a mouse or finger. Prefer the keyboard? Each point also has a “Position of point” slider you can move with the arrow keys.
3. Press ▶ and one point travels along the circle by itself. The measured angles change, but the “Check” row keeps its ✓ the whole time: that unchanging tick is the theorem.
4. Turn on “Show the construction lines of the proof” to see, as grey dashes, the extra lines a textbook proof draws: the diameter through C, the radii OA and OB, the line OP.
5. Change the radius and watch lengths rescale while every angle stays put. In tangent mode, drag P (or use the d slider) and compare the tangent length with √(d² − r²).

Bangles, clocks and bicycle wheels: why circles have rules

A bangle, a clock face, a bike wheel, a pizza, the ripple from a stone dropped in a pond: circles are everywhere. They all share one simple property. Every point on a circle is exactly the same distance from its centre. That single fact is enough to force a whole family of angle and length rules, and those rules are what we call the circle theorems.

Here is a football version. Imagine a circle that passes through both goalposts. Stand anywhere on that circle (on the same side as the pitch) and the angle between the two posts, the “shooting window”, is exactly the same size. Sports analysts use this to map where a shot has the widest view of goal. In maths language: angles in the same segment are equal.

This page covers seven theorems, and the simulation has a separate diagram for each. Drag the points however you like; the numbers change, but the relationship never breaks. Seeing that with your own eyes makes the proofs much easier to remember and much easier to write in an exam.

Circle vocabulary: every word you need, from centre to tangent

Most lost marks in circle questions come from mixing up two words. A chord is a straight line segment joining two points on the circle; an arc is a curved piece of the circle itself. A chord cuts the circle into two segments: the bigger one is the major segment and the smaller one is the minor segment. A sector is the “slice of pizza” bounded by two radii and an arc.

The table below gives each term in plain words, with where you can find it in the simulation.

Theorem 1: the angle at the centre is twice the angle at the circumference

The simulation opens on this theorem. The starting figure has radius 5, the angle at the centre on arc AB is ∠AOB = 120°, and the angle at the circumference is ∠ACB = 60°. Move C anywhere on the circle outside arc AB and ∠ACB does not change, while ∠AOB is always exactly double it.

Now drag A and B to make the arc bigger. At some point ∠AOB passes 180° and becomes a reflex angle. The rule still holds: ∠ACB turns obtuse, and twice it equals the reflex angle at the centre.

Theorem 2: angles in the same segment are equal

Chord AB splits the circle into two segments. From any two points C and D in the same segment, AB is seen under the same angle. In the starting figure ∠ACB = ∠ADB = 70°, because both stand on the same 140° arc.

Drag D across the chord to the other side and the two angles stop being equal; instead they add up to 180°. The readings panel switches to that relationship automatically, and it is exactly the cyclic quadrilateral theorem in disguise.

Theorem 3: the angle in a semicircle is a right angle

When the chord is a diameter, the angle at the centre is a straight angle, 180°. By Theorem 1 the angle at the circumference is half of that: 90°. So joining the ends of a diameter to any point on the circle always makes a right-angled triangle.

Choose “Semicircle” in the simulation. Dragging A rotates the whole diameter AB; dragging C slides the third corner. Either way a small square stays at C and ∠ACB reads 90°. Watch AC² + BC² and AB² too: Pythagoras is hiding in this diagram.

Theorem 4: opposite angles of a cyclic quadrilateral add up to 180°

If all four corners of a quadrilateral lie on a circle, each pair of opposite angles is supplementary. In the simulation’s starting quadrilateral ∠A = 100°, ∠B = 95°, ∠C = 80° and ∠D = 85°, so ∠A + ∠C = 180° and ∠B + ∠D = 180°.

A bonus rule comes free: extend BC to E and the exterior angle ∠DCE = 180° − ∠C = 100°, which equals the interior opposite angle ∠A. The green dashed line and green angle show it.

Theorem 5: the perpendicular from the centre bisects the chord

Drop a perpendicular OM from the centre to a chord AB and M lands exactly in the middle: AM = MB. In the simulation pick “Perpendicular to a chord” and drag A or B; the equal-length tick marks and the readings AM and MB always agree.

This theorem powers the most common circle calculation. Triangle OAM is right-angled with hypotenuse OA = r, so OM² + AM² = r², and the chord is AB = 2AM = 2√(r² − OM²).

Theorem 6: a tangent is perpendicular to the radius, and tangents from a point are equal

A tangent touches the circle at one point, and the radius to that point meets it at exactly 90°. From an outside point P two tangents can be drawn, and they are equal in length: PT₁ = PT₂ = √(d² − r²), where d = OP. At the start of the simulation r = 5 and d = 13, so each tangent is √(169 − 25) = √144 = 12.

Drag P closer and the tangents shrink; bring it right up to the circle and the two contact points almost merge. Pull it away and the tangents grow. They always stay equal, and both contact angles stay 90°.

Theorem 7: the alternate segment theorem

Draw a tangent at A and a chord AB from the point of contact. The angle between the tangent and the chord equals any angle at the circumference in the segment on the other side of the chord, the alternate segment. In the starting figure both read 60°.

“Alternate” means opposite: C sits in the segment on the far side of AB from the tangent angle. Move C and ∠ACB does not change; move A or B and both angles change together, always equal. This theorem is a favourite in GCSE, IGCSE and O level papers, often hidden inside a longer question.

Try it yourself: six quick experiments

Each takes a minute or two. Jot down the numbers you see; they make the proofs above feel obvious.

Worked examples, step by step

Every number below is calculated from the formula, not typed in. In an exam, write the reason for each step next to it; the reasons carry marks.

Common mistakes that cost marks

Examiners see these again and again. Read them once and you will spot them in your own work.

• Writing the angle at the centre as half the angle at the circumference. It is the other way round: the centre angle is the bigger one.
• Pairing angles that stand on different arcs. The centre angle and circumference angle must stand on the same arc.
• Treating angles on opposite sides of a chord as equal. They are supplementary, not equal.
• Adding adjacent angles of a cyclic quadrilateral to 180°. Only opposite angles do that.
• Finding half the chord with Pythagoras and forgetting to double it at the end.
• Putting the right angle at the external point P. The 90° is at the point of contact, between radius and tangent.
• Giving no reasons. “∠ACB = 90°” earns less than “∠ACB = 90° (angle in a semicircle)”.
• Using the wrong segment in the alternate segment theorem. The tangent–chord angle equals the angle in the segment on the other side.

Circle theorems in real life

A carpenter with a broken piece of a wheel can find the centre of the whole wheel: draw two chords on the piece and construct their perpendicular bisectors. They cross at the centre, because the perpendicular from the centre bisects a chord. Archaeologists size up whole plates from pottery shards the same way.

A set square finds the centre of any round object using the angle in a semicircle: put the right-angle corner on the rim, and the two points where its edges cross the rim are the ends of a diameter. Do it twice and the diameters cross at the centre.

Bicycle chains, belts and pulleys run along tangents. At the point where the belt leaves the wheel, the spoke (a radius) is perpendicular to the belt, and engineers use the tangent-length formula to work out belt lengths.

Road and railway curves are set out from a chord and the height of the arc above its midpoint, using OM² + AM² = r² to recover the radius. Theatre and cinema seats arranged on an arc through the screen edges give every seat the same viewing angle.

Exam corner: how circle theorems are tested

Circle theorems appear in GCSE and IGCSE Higher papers, O level, the SAT (angles and arcs), and every national curriculum’s Grade 9–10 geometry. Questions usually give a diagram with one or two angles and ask for another, with “give reasons for your answer”. The reason is often worth as much as the number.

Learn the standard wording of each reason, because mark schemes look for it: “angle at the centre is twice the angle at the circumference”, “angles in the same segment are equal”, “angle in a semicircle is 90°”, “opposite angles of a cyclic quadrilateral add up to 180°”, “tangent is perpendicular to the radius”, “tangents from an external point are equal”, “alternate segment theorem”.

• Mark every radius as equal on the diagram before you start; isosceles triangles hide in almost every question.
• Look for a diameter: it instantly gives you a right angle.
• Look for four points on the circle: they form a cyclic quadrilateral.
• Look for a tangent: mark the 90° at the point of contact straight away.
• For proof questions, write Given, To prove, Construction and Proof as separate headings, and a reason on every line.

One-screen revision summary

Glance at this the night before the exam. Every row matches one mode of the simulation.

Frequently asked questions