Perimeter of a Sector

Statement

The perimeter of a sector is the sum of the arc length and the two radii. It is given by:

\[ P = \frac{\theta}{360^\circ} \cdot 2\pi r + 2r \]

Why it’s true

• The perimeter of a sector includes two straight sides (both radii = \(2r\)) and a curved arc.
• The arc length is a fraction of the full circumference (\(2\pi r\)), proportional to \(\frac{\theta}{360}\).
• Adding them gives: \(P = \text{arc length} + 2r\).

Recipe (how to use it)

1. Identify the angle \(\theta\) and the radius \(r\).
2. Compute the arc length: \(\frac{\theta}{360} \times 2\pi r\).
3. Add \(2r\) for the two radii.

Spotting it

This formula is used when asked for the “perimeter” or “boundary length” of a sector (not just the arc).

Common pairings

• Problems about slices of pizza or cake (sector shapes).
• Geometry questions on circle sectors.
• Comparing perimeters and areas of sectors.

Mini examples

1. Given: Circle radius 6 cm, angle \(90^\circ\). Answer: \(P = (90/360)(2\pi \cdot 6) + 12 = 9.42 + 12 = 21.42\).
2. Given: Circle radius 10 cm, angle \(60^\circ\). Answer: \(P = (60/360)(2\pi \cdot 10) + 20 = 10.47 + 20 = 30.47\).

Pitfalls

• Forgetting to include the two radii (\(+2r\)).
• Using radians instead of degrees without converting.
• Mixing up perimeter with arc length (which is only the curved part).

Exam strategy

• Underline “perimeter” in the question to remind yourself it includes both radii.
• Write arc length separately, then add \(2r\).
• Round to 2 decimal places unless told otherwise.

Summary

The perimeter of a sector is the arc length plus the two radii. Use the formula: \(\;P = \frac{\theta}{360}\cdot 2\pi r + 2r\).