Arc & sector of a circle

A whole circle has a circumference and an area, and you already know the two formulas for them. But you almost never need the whole circle: a slice of cake, a piece of pizza, the patch a windscreen wiper cleans, a roundabout you leave again after a quarter turn. All of these shapes are parts of a circle, and for them you need no new formula, only a share.

The angle gives the share

A full circle has 360°. If your slice has a central angle of 90°, it is a quarter of the circle, because $90 : 360 = 0.25$. At 60° it is a sixth, at 120° a third. This fraction $\frac{\alpha}{360°}$ is the share, and you simply multiply it by whatever you know about the whole circle.

Arc, sector and the perimeter of the slice

This is where the most common mistake happens, so a clear separation is worth it. The arc is only the round edge of the slice. The sector is the area of the slice. But if you want the perimeter of a slice of cake, so the way all the way round it, then the two straight edges join the arc, and each of them is $r$ long. The perimeter of the sector is therefore $b + 2r$.

An example with numbers: $r = 6$ cm and $\alpha = 60°$. The share is $\frac{60}{360} = \frac{1}{6}$. The whole circumference is $2 \pi \cdot 6 \approx 37.7$ cm, so the arc is about 6.3 cm. The whole area is $\pi \cdot 36 \approx 113.1$ cm², so the sector is about 18.8 cm². And the perimeter of the slice is $6.3 + 12 = 18.3$ cm.