The perimeter of a circular sector is its arc length plus the two radii along its straight edges. If the angle is in radians, multiply the radius by the angle to find the arc. If the angle is in degrees, convert it to radians first or use the matching fraction of the full circumference.
The 2027 SEC G3 Mathematics K310 syllabus covers arc length, sector area and radian measure. The examples below compare a minor sector and a major sector from the same circle, then show how to work backwards from a perimeter.
- Trace the boundary of the shaded sector.
- Check whether the angle belongs to the minor or major arc.
- Use radians in the formula s = rθ.
- Add both straight edges when finding the sector perimeter.
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Trace the shape before choosing the formula
Start at the centre, move out along one radius, follow the arc around the shaded region and return along the other radius. Those three pieces make the sector boundary. The chord between the ends of the arc belongs to a circular segment boundary instead.
An angle of θ degrees covers θ/360 of the circumference. An angle of θ radians gives s = rθ directly. For example, 120° is 2π/3 radians, so the two routes give the same arc length.
A major sector goes around more than half the circle. If the given angle describes the smaller unshaded sector, use the remaining angle for the shaded arc.
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Worked example | The minor and major sectors
A circle has radius 9 cm. Two radii form a smaller central angle of 120°. Find the perimeters of the minor sector and the major sector.
| Sector | Central angle | Arc length | Perimeter |
|---|---|---|---|
| Minor | 120° | 6π cm | 18 + 6π cm |
| Major | 240° | 12π cm | 18 + 12π cm |
The minor arc is one third of the full circumference of 18π cm, so it measures 6π cm. The major arc is the other two thirds, so it measures 12π cm. Add 9 cm twice to each arc length.
As a check, the two arcs add to 18π cm. The two sector perimeters add to 36 + 18π cm because both straight edges are counted in each perimeter.
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CHAPTER 3 OF 4 · Back to contents
Work backwards from the perimeter
A sector has radius 8 cm and perimeter 22 cm. The straight edges account for 16 cm, so the arc length is 22 − 16 = 6 cm. Then θ = s/r = 6/8 = 0.75 radians.
The full circle is 2π radians, so 0.75 radians is a plausible angle for a small sector. If asked for degrees, multiply by 180/π to obtain about 43.0°.
In a perimeter question, write the units next to the arc and radii. Area formulae give square units and describe the space inside the boundary.
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CHAPTER 4 OF 4 · Back to contents
Frequently asked questions
Should I include the chord?
Include the chord if the boundary being measured consists of a chord and an arc. A sector uses two radii as its straight edges.
When should I round π?
Keep π in the working when possible. Give an exact answer or round the final value as requested.
What if the sector is a semicircle?
Its two radii together make the diameter. The perimeter is half the circumference plus that diameter.
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