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G3.16 Circle Area and Circumference

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01 — Explanation

What's the relationship between radius, diameter, and \pi?

The diameter is twice the radius: $d=2r$ . Pi ( $\pi\approx3.14159$ ) is the fixed ratio of circumference to diameter for every circle. Both formulas are given on the exam formula sheet:

C = \pi d = 2\pi r \qquad\qquad A = \pi r^2

Exam Tip: Use your calculator's $\pi$ button throughout, and only round the final answer — usually to 3 significant figures unless told otherwise.

02 — Explanation

How do I find the circumference?

If you're given the radius, double it first to get the diameter, then multiply by $\pi$ .

Example: radius $=6.2$ m. Diameter $=12.4$ m. $C=\pi\times12.4\approx38.96$ , so $C \approx 39.0$ m (3 s.f.).

03 — Explanation

How do I find the area?

If you're given the diameter, halve it first to get the radius — the area formula always needs the radius, never the diameter.

Example: diameter $=18$ cm. Radius $=9$ cm. $A=\pi\times9^2=81\pi\approx254.47$ , so $A\approx254$ cm² (3 s.f.).

Common Misconception: Substituting the diameter directly into $A=\pi r^2$ instead of halving it first — this gives an area four times too big, since you've effectively squared $d$ instead of $r$ . Always write down $r=d\div2$ as its own explicit step.

04 — Explanation

How do I convert an area between cm² and m²?

The linear factor must be squared for area — $1\text{ m}=100\text{ cm}$ means $1\text{ m}^2 = 100^2\text{ cm}^2=10{,}000\text{ cm}^2$ , not just $\times100$ .

Exam Tip: To sidestep this entirely, convert the radius or diameter into your target unit before using the area formula — e.g. convert $9$ cm to $0.09$ m first, then $A=\pi(0.09)^2$ comes out in m² automatically.

05 — Explanation

How do I find the area or perimeter of a semicircle or quarter circle?

For area, take the matching fraction of the full circle's area: semicircle $=\frac12\pi r^2$ ; quarter circle $=\frac14\pi r^2$ ; for a general sector of angle $\theta$ , use $\frac{\theta}{360}\times\pi r^2$ .

For perimeter, you need the curved arc plus the straight edge(s) that close the shape — not just the arc on its own.

\text{Semicircle perimeter} = \pi r + 2r = r(\pi+2)

Common Misconception: The "diameter trap" — calculating a semicircle's perimeter as just half the circumference ( $\pi r$ ) and forgetting to add the straight diameter edge that seals the shape. Always sketch and label which parts are curved and which are straight before adding them up.

06 — Explanation

How do I find the radius if I'm only given the area or circumference?

Rearrange the formula before substituting — treat $A=\pi r^2$ or $C=2\pi r$ as an equation to solve for $r$ .

Example: A circle has circumference $50$ cm. Find its radius.

$50 = 2\pi r$ .
Divide both sides by $2\pi$ : $r = \dfrac{50}{2\pi} \approx 7.96$ cm (3 s.f.).

For area, the same idea applies but needs a square root at the end: if $A=\pi r^2$ , then $r=\sqrt{A\div\pi}$ .

Exam Tip: Keep $\pi$ as a symbol for as long as possible in these "reverse" problems — dividing by $2\pi$ directly on your calculator in one step avoids an extra rounding error from evaluating $\pi$ too early.

07 — Worked example

Worked Example

A semicircular flowerbed has a diameter of $14$ m. Find (a) its area and (b) its perimeter, both to 3 s.f.

Radius: $14 \div 2 = 7$ m.
(a) Area: $\frac12 \pi r^2 = \frac12 \times \pi \times 7^2 = \frac12\times49\pi \approx 76.97$ , so $76.9$ m² (3 s.f.).
(b) Perimeter: arc $=\pi r \approx 21.99$ m, plus the straight diameter $=14$ m, giving $21.99+14=35.99$ , so $36.0$ m (3 s.f.).