What Is the Perimeter of a Sector?
A sector is the "pizza slice" region of a circle bounded by two radii and the arc between them. Its perimeter is the total distance around that slice: the two straight radii plus the curved arc. This calculator computes that perimeter instantly from the radius and the central angle, in either degrees or radians.
How to Use This Calculator
Enter the radius (\(r\)) of the circle and the central angle of the sector. Choose whether the angle is in degrees or radians using the dropdown, then read off the perimeter. The tool also shows the arc length and the angle converted to radians so you can see each part of the result.
The Formula Explained
The perimeter is given by $$P = 2r + r\theta$$ where \(\theta\) is the central angle in radians. The term \(r\theta\) is the arc length. If your angle is in degrees, convert it first using \(\theta = \pi \times \text{degrees} \div 180\), which gives the degree form $$P = 2r + \frac{\pi \cdot r \cdot \text{degrees}}{180}$$ The "\(2r\)" accounts for the two straight edges of the slice.
Worked Example
Suppose \(r = 5\) and the central angle is \(60°\). Convert: $$\theta = \pi \times 60 \div 180 = 1.04720 \text{ rad}$$ Arc length \(= 5 \times 1.04720 = 5.23599\). Perimeter \(= 2 \times 5 + 5.23599 =\) 15.23599 units.
FAQ
Does the angle have to be in radians? No. Pick degrees or radians from the dropdown; the calculator converts as needed.
What if the angle is 360°? The arc becomes the full circumference (\(2\pi r\)), so the perimeter is \(2r + 2\pi r\) — the radii plus the whole circle edge.
What units does the perimeter use? The same length unit as the radius. If \(r\) is in cm, the perimeter is in cm.
