MISSION · Geometry
Circle Theorems
A handful of facts that make circle diagrams solvable
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LEVEL 2 · LEARN
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The idea in one sentence
A few facts about angles in circles turn diagrams that look unsolvable into one-line subtractions.
The angle at the centre
The angle at the centre is twice the angle at the circumference, when both stand on the same arc.
Two points on a circle, joined to the centre, make one angle. Join the same two points to any other point on the circle and you get an angle exactly half the size. Move that third point anywhere on the major arc and the angle doesn’t change — which is the second theorem hiding inside the first.
Angles in the same segment are equal. Every viewing point on the same arc sees the chord at the same angle. It’s the reason a football pitch’s penalty arc is shaped as it is: every point on it sees the goal at the same angle.
The angle in a semicircle
If the two points are the ends of a diameter, the central angle is a straight \(180°\), so the angle at the circumference is \(90°\). Always.
Any triangle drawn on a diameter, with its third corner on the circle, is right-angled. This is Thales’ theorem, and it’s the easiest one to spot in an exam: look for a diameter and a point on the circle, and you’ve found a right angle without being told.
Cyclic quadrilaterals
A quadrilateral with all four corners on a circle has opposite angles adding to \(180°\).
It follows from the centre theorem. The two opposite angles stand on the two arcs that together make the whole circle, so their doubles total \(360°\), so they total \(180°\).
Tangents
A tangent touches the circle at exactly one point, and it is perpendicular to the radius at that point. Every tangent question starts there — draw the radius to the point of contact and a right angle appears.
Two more, both worth knowing:
- Two tangents from the same external point are equal in length, so the figure they make with the two radii is symmetrical.
- The alternate segment theorem: the angle between a tangent and a chord equals the angle in the opposite segment. This is the one people forget, and it’s usually the one the hard question needs.
How to use them
Circle problems are rarely about calculation. They’re about spotting which theorem the diagram is set up for, and the setup is always visible:
- A diameter drawn in? Look for the right angle.
- Lines to the centre? Central and circumference angles.
- Four points on the circle? Cyclic quadrilateral.
- A line touching once? Tangent, so draw the radius.
Then write the reason alongside the number. In most exams the reason carries as many marks as the answer.
Common mix-ups
- Halving the wrong one. The centre angle is the bigger one, twice the circumference angle.
- Using the theorems on angles standing on different arcs. They must share the same two endpoints.
- Assuming a chord through the middle-ish region is a diameter. It has to pass through the centre.
- Giving an answer with no reason, which loses marks even when the number is right.
Check yourself
A triangle is drawn with one side as the circle’s diameter and its third corner on the circle. One of the other angles is \(35°\). Find the remaining two angles.
The angle at the circumference is \(90°\) by the semicircle theorem, so the third is \(180 - 90 - 35 = 55°\).
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