Perimeter and Circumference

So a circle with radius $7\text{ cm}$ has diameter $14\text{ cm}$ and circumference $\pi \times 14 \approx 44.0\text{ cm}$.

You can also use the formula in reverse. If you know the circumference, the diameter is $d = \dfrac{C}{\pi}$, and halving that gives the radius.

This year you push further: compound shapes that mix straight and curved edges, the perimeter of a sector, and multi-step problems where arc length is one part of a longer calculation. The first habit, every time, is to decide whether you have been given the radius or the diameter.

• Circumference is a circle's perimeter: C = π × d.
• It can also be written C = 2 × π × r.
• Rearranged, the diameter is d = C ÷ π.
• Always check whether you have the radius or the diameter.

For a circle of radius $9\text{ cm}$ with a $120\degree$ sector:

1. Full circumference: $C = 2 \times \pi \times 9 \approx 56.5\text{ cm}$.
2. The fraction is $\dfrac{120}{360} = \dfrac{1}{3}$.
3. Arc length $= \dfrac{1}{3} \times 56.5 \approx 18.8\text{ cm}$.

A quick check: an angle of $180\degree$ gives the fraction $\tfrac{1}{2}$ — the curved part of a semicircle. An angle of $90\degree$ gives a quarter circle. The fraction always matches the slice you can picture.

• An arc is part of the circumference of a circle.
• Arc length = (angle ÷ 360°) × circumference.
• A 180° arc is half the circumference; a 90° arc is a quarter.
• Find the full circumference first, then take the fraction.

Example: a sector has radius $10\text{ cm}$ and angle $90\degree$.

1. Full circumference: $C = 2 \times \pi \times 10 \approx 62.8\text{ cm}$.
2. Arc length: $\dfrac{90}{360} \times 62.8 \approx 15.7\text{ cm}$.
3. Add the two radii: $15.7 + 10 + 10 = 35.7\text{ cm}$.

The same care applies to a semicircle, which is just a $180\degree$ sector: its perimeter is the curved half plus the straight diameter.

• A sector has three edges: an arc and two radii.
• Sector perimeter = arc length + 2 × radius.
• Trace the whole boundary so no edge is missed.
• A semicircle is a 180° sector — add the diameter too.

• Three straight sides: $20 + 20 + 12 = 52\text{ cm}$.
• The curved end is a semicircle of diameter $12\text{ cm}$: $\dfrac{\pi \times 12}{2} \approx 18.8\text{ cm}$.
• Total perimeter $\approx 52 + 18.8 = 70.8\text{ cm}$.

The straight side that the semicircle replaces is not part of the boundary, so it is not counted.

• Split the boundary into straight parts and curved parts.
• Find each straight part and each curved part separately.
• Add only the outside edges — never an internal line.
• A side replaced by a curve is not part of the perimeter.

From an arc length to an angle. Rearrange the arc formula. Since $\text{arc} = \dfrac{\text{angle}}{360\degree} \times C$: $\text{angle} = \frac{\text{arc length}}{C} \times 360\degree.$

Example: a sector has radius $6\text{ cm}$ and an arc of $9.42\text{ cm}$.

1. Full circumference: $C = 2 \times \pi \times 6 \approx 37.7\text{ cm}$.
2. Angle $= \dfrac{9.42}{37.7} \times 360\degree \approx 90\degree$.

So the sector is a quarter circle. Working backwards is just careful rearranging — decide what you know, what you want, and which formula links them.

• From a circumference: d = C ÷ π, then halve for the radius.
• From an arc length: angle = (arc ÷ C) × 360°.
• Find the full circumference first as your stepping stone.
• Decide what you know and what you want before rearranging.