Circumference and area of a circle, arc length and sector area, and the circle theorems at Higher tier.

What this dot point is asking

AQA wants you to find the circumference and area of a circle, the length of an arc and the area of a sector, and at Higher tier to apply the circle theorems with reasons. Circle questions appear at both tiers, and the Higher circle theorems are a reliable source of "state the reason" marks, so learning the theorem names precisely matters.

Circumference and area

A circle of radius $5\,\text{cm}$ has circumference $2\pi \times 5 = 10\pi \approx 31.4\,\text{cm}$ and area $\pi \times 5^2 = 25\pi \approx 78.5\,\text{cm}^2$. To work backward, a circle of area $36\pi\,\text{cm}^2$ has $r^2 = 36$, so $r = 6\,\text{cm}$. Keep answers in terms of $\pi$ when the question asks, otherwise round as instructed.

Arc length and sector area

An arc is part of the circumference and a sector is a pie-slice of the area. Both are the same fraction $\dfrac{\theta}{360}$ of the whole circle, where $\theta$ is the angle at the centre in degrees.

Circle theorems at Higher tier

The circle theorems give angle relationships for points on a circle. Learn each name as the reason:

• Angle at the centre is twice the angle at the circumference (subtended by the same arc).
• Angle in a semicircle is $90^\circ$ (a special case where the chord is a diameter).
• Angles in the same segment are equal (subtended by the same arc).
• Opposite angles of a cyclic quadrilateral sum to $180^\circ$.
• A tangent is perpendicular to the radius at the point of contact.
• Tangents from an external point are equal in length, and the alternate segment theorem links a tangent-chord angle to the angle in the alternate segment.

For example, if the angle at the centre subtended by an arc is $130^\circ$, the angle at the circumference subtended by the same arc is $65^\circ$, because the centre angle is twice the circumference angle.

Chaining circle theorems with reasons

Higher questions often need two or three theorems in sequence, each justified. A typical chain: a cyclic quadrilateral has one angle of $95^\circ$, so the opposite angle is $180^\circ - 95^\circ = 85^\circ$ (opposite angles of a cyclic quadrilateral sum to $180^\circ$); then if a radius meets a tangent at a labelled vertex, that angle is $90^\circ$ (tangent perpendicular to radius). Write each step with its reason on the same line, because the reasoning marks are awarded per theorem used. Building the answer one justified step at a time, rather than jumping to the final number, is what secures full marks.

Working in terms of pi and exact answers

Many circle questions ask for answers "in terms of $\pi$", meaning you leave $\pi$ as a symbol rather than evaluating it. A circle of radius $7\,\text{cm}$ has area $49\pi\,\text{cm}^2$ and circumference $14\pi\,\text{cm}$ exactly. Leaving $\pi$ in keeps the answer precise and is often quicker on a non-calculator paper. Only replace $\pi$ with $3.14159\ldots$ when the question asks for a decimal or a rounded value, and then round at the very end to avoid accumulating rounding errors through the calculation.

Segments and the area between shapes

A segment is the region between a chord and the arc it cuts off, found by subtracting the area of the triangle from the area of the sector. Composite area questions, such as the area inside a square but outside an inscribed circle, are solved by finding each area and subtracting. These combine the circle formulae with the area work from other shapes, so a clear plan (which areas to add, which to subtract) is the key to not losing track.