Parts of a circle, arc length and sector area, and the circle theorems including the angle at the centre, angles in a semicircle, angles in the same segment, cyclic quadrilaterals, the tangent-radius angle and the alternate segment theorem (Higher tier).

What this dot point is asking

Edexcel expects you to know the parts of a circle, to find arc lengths and sector areas as fractions of the whole circle, and at Higher tier to apply the circle theorems to find angles, always giving the named theorem as the reason. Circle theorems are a classic Higher topic and reward recognising which theorem a diagram is testing.

Parts of a circle

Knowing the vocabulary is the first step, because theorems are stated in these terms.

Arc length and sector area

An arc and a sector are simply fractions of the whole circle, with the fraction equal to the angle over $360^\circ$.

So a sector of radius $6\,\text{cm}$ and angle $90^\circ$ has arc length $\tfrac{90}{360} \times 2\pi \times 6 = 3\pi\,\text{cm}$ and area $\tfrac{90}{360} \times \pi \times 36 = 9\pi\,\text{cm}^2$. The perimeter of a sector includes the two radii as well as the arc.

The circle theorems (Higher)

The circle theorems let you find angles inside circles. Each has a precise reason that must be quoted.

Many circle-theorem questions chain two or three theorems together, so identify each angle's theorem one at a time. Always finish by naming the theorem, because that is where the reasoning marks are.

Working through a chained diagram

A typical Higher question marks several angles on one diagram and asks you to reach a final angle, justifying each step. The strategy is to write down every angle you can find immediately, each with its reason, then use those to unlock the next. For example, if a triangle has one vertex at the centre and the radius meets a tangent, you might first use "tangent meets radius at $90^\circ$", then "angles in a triangle sum to $180^\circ$", then "the angle at the centre is twice the angle at the circumference". Setting out the reasons in order shows the examiner a logical chain and secures every reasoning mark, even if one arithmetic step slips.

Try this

Q1. A sector has radius $12\,\text{cm}$ and angle $60^\circ$. Work out the arc length in terms of $\pi$. [2 marks]

• Cue. $\dfrac{60}{360} \times 2\pi \times 12 = \dfrac{1}{6} \times 24\pi = 4\pi\,\text{cm}$.

Q2. $P$ and $Q$ are points on a circle and $PQ$ is a diameter. $R$ is a third point on the circle. What is the size of angle $PRQ$, and why? [2 marks]

• Cue. $90^\circ$, because the angle in a semicircle is a right angle.