Study notes

Arc Length & Sector Area

Finding arc length and sector area as a fraction of the full circle, working backwards to find a missing radius or angle, and sector perimeters, all in degrees with pi = 3.14.

Learn it step by step

A sector is a fraction of a full circle

A sector's angle $θ$ (at the centre) is some fraction of the full $360°$ turn: that fraction is $\frac{θ}{360}$ . Both the arc length and the sector area are found by multiplying this same fraction by the corresponding full-circle formula.

A sector's angle theta out of 360 degrees gives the fraction of the circle it covers.

Arc length: a fraction of the circumference

Arc length $= \frac{θ}{360} \times 2 π r$ , since the full circumference is $2 π r$ . Use degrees throughout and $π = 3.14$ for numeric answers, never radians in this course.

Sector area: a fraction of the circle's area

Sector area $= \frac{θ}{360} \times π r^{2}$ , since the full circle's area is $π r^{2}$ . A common slip is using $r$ instead of $r^{2}$ , or mixing up this formula with the arc length formula.

Working backwards to find the radius or the angle

If the arc length or sector area is already given, rearrange the formula instead of guessing: from $arc = \frac{θ}{360} \times 2 π r$ , either $r = \frac{arc \times 360}{2 π θ}$ (if $θ$ is known) or $θ = \frac{arc \times 360}{2 π r}$ (if $r$ is known). The same idea applies to sector area with $π r^{2}$ in place of $2 π r$ .

The perimeter of a sector is the arc plus two radii

A sector's boundary has three parts: the curved arc, and TWO straight radii joining the centre to each end of the arc. So $perimeter = arc length + 2 r$ . A common mistake is giving only the arc length as the perimeter, or adding just one radius.

Worked examples

Find the arc length of a sector with radius 10cm and angle 90°. Use pi = 3.14.

Arc length $= \frac{θ}{360} \times 2 π r = \frac{90}{360} \times 2 \times 3.14 \times 10$ .
$\frac{90}{360} = 0.25$ , and $2 \times 3.14 \times 10 = 62.8$ .
Arc length $= 0.25 \times 62.8 = 15.7$ cm.

A sector has an area of 47.1cm squared and a radius of 10cm. Find the angle theta. Use pi = 3.14.

Sector area $= \frac{θ}{360} \times π r^{2}$ , so $47.1 = \frac{θ}{360} \times 3.14 \times 100 = \frac{θ}{360} \times 314$ .
Rearrange: $θ = \frac{47.1 \times 360}{314}$ .
$θ = \frac{16956}{314} = 54°$ .

Find the perimeter of a sector with radius 8cm and angle 45°. Use pi = 3.14.

First find the arc length: $\frac{45}{360} \times 2 \times 3.14 \times 8 = 0.125 \times 50.24 = 6.28$ cm.
The perimeter adds the two straight radii: perimeter $= arc + 2 r = 6.28 + 2 (8)$ .
Perimeter $= 6.28 + 16 = 22.28$ cm.

A sector OAB has radius 12cm and angle 150°. Find the arc length AB, the perimeter of the sector, and its area. Use pi = 3.14, rounding each to 2 decimal places.

Arc length: $\frac{150}{360} \times 2 \times 3.14 \times 12 = 0.41 6 \times 75.36 = 31.40$ cm.
Perimeter: arc plus two radii, $31.40 + 2 (12) = 31.40 + 24 = 55.40$ cm.
Area: $\frac{150}{360} \times 3.14 \times 144 = 0.41 6 \times 452.16 = 188.40$ cm squared.

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