Introduction to Circle Theorems

The parts of a circle

Before the theorems, learn the vocabulary — every rule is stated using these words:

• Centre — the middle point.
• Radius — a line from the centre to the edge.
• Diameter — a line through the centre joining two edges (twice the radius).
• Chord — a straight line joining two points on the circle (a diameter is the longest chord).
• Arc — part of the curved edge.
• Tangent — a straight line that just touches the circle at one point.
• Sector — a "pizza slice" between two radii.
• Segment — the region between a chord and an arc.

The word subtend matters: an angle is subtended by an arc when its two arms reach the ends of that arc.

Theorem 1: angle in a semicircle

The angle in a semicircle is 90°.

If a triangle is drawn inside a circle so that one side is a diameter, then the angle opposite the diameter — the one touching the circle — is always a right angle.

Described diagram: picture a circle with a horizontal diameter across the middle. A point sits on the top of the circle, joined to both ends of the diameter. The triangle formed has a perfect right angle at that top point, no matter where on the arc the point sits.

Worked example: a triangle inside a circle has the diameter as its base, and one base angle is 35°. Find the other two angles.

• The angle opposite the diameter = 90° (angle in a semicircle).
• Angles in a triangle sum to 180°, so the last angle = 180 − 90 − 35 = 55°.

Theorem 2: angle at the centre

The angle at the centre is twice the angle at the circumference when both stand on the same arc.

Two angles are drawn on the same arc: one with its point at the centre, one with its point on the circumference. The centre angle is always double the circumference angle.

Worked example: the angle at the centre on a particular arc is 100°. The angle at the circumference on the same arc = 100 ÷ 2 = 50°.

Theorem 3: angles in the same segment

Angles in the same segment are equal.

If several angles stand on the same arc and have their points in the same segment, they are all equal to each other. This actually follows from Theorem 2 — each circumference angle is half the same centre angle, so they must match.

Worked example: two angles both stand on the same arc in the same segment, and one is 38°. The other is also 38°.

Theorem 4: cyclic quadrilaterals

A cyclic quadrilateral has all four corners on the circle.

Opposite angles of a cyclic quadrilateral add up to 180°.

Worked example: a cyclic quadrilateral has angles 95°, 80°, x and y, where x is opposite 95° and y is opposite 80°.

• x = 180 − 95 = 85°
• y = 180 − 80 = 100°

Always give a reason

Circle theorem problems are marked on reasoning, not just answers. For every step, name the theorem you used, for example: "x = 90° (angle in a semicircle)". A correct number with no reason often scores zero.

Activity: discover the centre rule

Draw a large circle and mark the centre. Choose an arc and draw the angle at the centre and the angle at the circumference standing on it. Measure both with a protractor. You should find the centre angle is twice the circumference one — proving Theorem 2 for yourself. Try a few different arcs to confirm it always holds.

Where this connects

These theorems extend the angle rules from angles and lines into curved geometry, and they build on knowing the parts of a circle from circles: circumference and area.