Circle Theorem Circus

Circle Theorem Circus 🎪

Welcome to the **Higher-tier** big top. Circle theorems look intimidating, but there are only a handful, and each one is a single, quotable fact. The winning routine every time: **spot** which theorem the diagram is showing, **apply** it to get the angle, and **name** it — a circle-theorem answer with no reason drops marks.

Know the parts of a circle ⭕

The theorems are built from these words, so lock them in first: • **Radius** — centre to the edge. • **Diameter** — a chord *through the centre* (2 radii). • **Chord** — a straight line joining two points on the circle. • **Tangent** — a line that touches the circle at exactly one point. • **Arc** — part of the circumference. • **Sector** — a 'pizza slice'. • **Segment** — the region a chord cuts off.

Tap the diameter

An interactive activity.

Match each part to its meaning

Centre is double 🎯

The first big theorem: the angle at the **centre** is **twice** the angle at the **circumference**, when both are drawn from the **same arc** (the same two points). So if the angle at the circumference is 25°, the angle at the centre is 50°. The **centre** is always the **bigger** one — double, never half.

Angle at the centre

An interactive activity.

Angle in a semicircle 📐

A neat special case: when the chord is a **diameter**, the angle it makes at the circumference is always **90°** — the *angle in a semicircle*. (It follows from the last theorem: the diameter is a straight angle of 180° at the centre, and half of 180° is 90°.) Spot a triangle with the diameter as one side — that corner is a right angle.

The semicircle angle

A triangle is drawn inside a circle so that one side is a diameter. What is the angle at the vertex on the circumference?

• 90°
• 45°
• 180°
• It depends on the triangle

Same segment and cyclic quadrilaterals 🟰

Two more to add to the act: • **Angles in the same segment are equal** — angles at the circumference standing on the same chord are the same size. • A **cyclic quadrilateral** has all four corners on the circle, and its **opposite angles add to 180°**.

Cyclic quadrilateral

An interactive activity.

Tangent theorems 📏

Finally, the tangents: • A **tangent meets a radius at 90°** at the point where it touches — a right angle, exactly. • **Two tangents drawn from the same external point are equal in length**. These often combine with the others in a multi-step chase, so name each one as you go.

Tangent meets radius

A tangent touches a circle, and the radius is drawn to the point of contact. What is the angle between the tangent and the radius?

• 90°
• 180°
• 45°
• It varies around the circle

Order the working

An interactive activity.

Quote the theorems

The angle at the centre is _____ the angle at the circumference. The angle in a semicircle is _____ degrees. A tangent meets a radius at a right angle, and opposite angles of a cyclic quadrilateral add to _____ degrees.