Which of these correctly describes a tangent to a circle?

It touches the circle at exactly one point

It is perpendicular to the radius at the point of contact

It is a chord that bisects the circle

Circles

⏱ 10 min✏️ Quiz at the end

Key Circle Vocabulary

Before applying formulas or theorems, it is essential to know the parts of a circle.

Term	Definition
Centre	The middle point of a circle, equidistant from all points on the circumference
Radius (r)	A straight line from the centre to the circumference
Diameter (d)	A chord passing through the centre; d = 2r
Circumference	The perimeter (total distance around) the circle
Chord	A straight line joining two points on the circumference
Arc	Part of the circumference between two points
Sector	A "pie slice" — the region between two radii and an arc
Segment	The region between a chord and the arc it cuts off
Tangent	A straight line that touches the circle at exactly one point

Circumference

The circumference is the distance all the way around a circle.

Formulas:

C = πd (using diameter)
C = 2πr (using radius)

Worked example: Find the circumference of a circle with diameter 14cm.

C = π × 14 ≈ 3.14 × 14 = 43.96cm

Worked example: Find the circumference with radius 9cm.

C = 2 × π × 9 ≈ 2 × 3.14159 × 9 ≈ 56.55cm

Use the π button on your calculator for more precise answers in exams.

Area of a Circle

Formula: A = πr²

Always use the radius, not the diameter. If given a diameter, halve it first.

Worked example: Find the area of a circle with radius 5cm.

A = π × 5² = π × 25 ≈ 3.14 × 25 = 78.5cm²

Worked example: Find the area of a circle with diameter 10cm.

Radius = 10 ÷ 2 = 5cm
A = π × 5² = 78.5cm² (same result)

Arc Length and Sector Area

A sector is a fraction of a circle determined by a central angle θ (theta).

Arc length (the curved edge of a sector):

Arc length = θ/360 × 2πr

Sector area:

Sector area = θ/360 × πr²

Worked example: Sector with radius 6cm and angle 90°.

Arc length = 90/360 × 2 × π × 6 = 1/4 × 12π ≈ 9.42cm
Sector area = 90/360 × π × 6² = 1/4 × 36π ≈ 28.27cm²

Think of it as: what fraction of a full circle is the sector? That same fraction applies to both arc length and area.

Circle Theorems

Circle theorems describe fixed angle relationships in circles. You must know these for GCSE and beyond. Always state the theorem name when giving reasons in an exam.

Theorem 1 — Angle at the centre: The angle subtended at the centre is twice the angle subtended at the circumference by the same arc.

Centre angle = 2 × circumference angle

Theorem 2 — Angle in a semicircle: The angle in a semicircle (standing on a diameter) is always 90°.

Theorem 3 — Angles in the same segment: Angles subtended by the same arc in the same segment are equal.

Theorem 4 — Cyclic quadrilateral: Opposite angles of a cyclic quadrilateral (all four vertices on the circumference) add up to 180°.

Theorem 5 — Tangent and radius: A tangent to a circle is perpendicular (90°) to the radius at the point of contact.

Theorem 6 — Two tangents from a point: Tangents drawn from an external point to a circle are equal in length.

Theorem 7 — Perpendicular from centre to chord: The perpendicular from the centre of a circle to a chord bisects (cuts in half) the chord.

Worked Example: Circle Theorems

Problem: O is the centre of a circle. Angle BOC = 110°. Find angle BAC where A is a point on the major arc.

Using Theorem 1:

Angle at circumference = 110° ÷ 2 = 55°

Problem: ABCD is a cyclic quadrilateral with angle A = 75° and angle C = ?

Using Theorem 4:

Angle C = 180° - 75° = 105°

Tangent Properties

A tangent touches the circle at exactly one point (the point of tangency). Key facts:

The tangent is perpendicular to the radius at the point of contact (90°)
Two tangents drawn from the same external point are equal in length
The angle between a tangent and a chord equals the angle in the alternate segment (tangent-chord angle theorem)

Common Mistakes

Using diameter instead of radius in the area formula — A = πr², not πd².
Confusing arc length and sector area — arc length uses 2πr; sector area uses πr².
Not citing the theorem — in exam answers, always name the circle theorem you are using.
Forgetting the factor of 2 in the angle-at-centre theorem — the centre angle is TWICE the circumference angle, not equal.
Mixing up segment and sector — a sector is a pizza slice; a segment is the region between a chord and the arc.

Tips and Tricks

Learn the sector formulas as fractions of the full circle: arc length = fraction × full circumference; sector area = fraction × full area.
For circle theorem questions, always draw the radius to the point of tangency to create the 90° angle — it often unlocks the whole problem.
Mark equal lengths (two radii) and equal angles on your diagram before searching for the unknown.
When stuck on a theorem question, list all the theorems you know and check which one applies to the configuration shown.