How do you find the arc length or perimeter of a circle sector?

Category: 2D Shapes »

Course: Geometry »

Relevant MM Lessons:
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Sector Arc Length and Perimeter

Circle sectors are partial circles. They are often smaller than a semi-circle and look like a "pizza slice" but can be bigger than a semi-circle and look like the remaining pizza after you remove a slice (which I affectionately refer to as a "pac-man").orTheir most important quality is the fact that they are a partial circle. If we know the circumference of the whole circle, we can find the arc curve of a fraction of it. And if we're asked for the entire perimeter of the sector, we will just add the radius lengths.A sector has the same radius length as the whole circle to which it belongs. We know the circumference of a whole circle is $2 \pi r$, so all we have to do is consider what fraction of the circle the sector represents, and then take that same fraction of the total circumference to get the curved arc length.The key lies in the central angle. If a sector has a central angle of $x$°, then the sector represents $x/360$ fraction of the whole circle. Indeed, we can see that if the central angle was the full $360$°, that the sector would be one whole circle.The arc length of a circle (the curved part of the sector) is given by the formula$$\mathrm{Arc} \,\,\,\, \mathrm{Length} \,\,\,\, = \frac{x}{360} \cdot 2\pi r$$where $x$ is the measure of the central angle in degrees, and $r$ is the radius of the circle to which the sector belongs. Conceptually, you should think of this formula as [fraction] $\times$ [whole circle circumference].This means that if you are asked for perimeter instead of arc length, they are asking you to also include the straight line parts of the sector, which are each length $r$.$$\mathrm{Perimeter} \,\,\,\, = \frac{x}{360} \cdot 2\pi r + 2r$$If you are measuring your central angle in radian measure, then we will ratio the central angle $\theta$ against $2\pi$ radians, since $360$° is equivalent to $2\pi$ radians.$$\mathrm{Arc} \,\,\,\, \mathrm{Length} \,\,\,\, = \frac{\theta}{2 \pi} \cdot 2\pi r$$Additionally, as a small shortcut, if you are measuring your central angle in radians, you can simplify the sector arc length expression (curved part only) to say$$\mathrm{Arc} \,\,\,\, \mathrm{Length} \,\,\,\, = r \theta$$where $\theta$ is the central angle of the sector measured in radians.