SSC CGL Maths · Mensuration 2D

Arc Length and Sector Perimeter

An arc is the same fraction of the circumference as its angle is of 360°, and a sector's perimeter is that arc plus two radii.

All Mensuration 2D notes

Why this matters

Nineteen questions in the bank, fifteen EASY and four MODERATE; ten come from the 2024 shifts and four from 2025. The angle fraction learnt here is the same one the sector-area page uses, and radians, which thirteen questions from the 2025 shifts use, are introduced on this page.

Concept 1 of 3: Arc length as a fraction of the circumference

Cut a round cake into pieces at the centre. A piece with a 90° corner carries a quarter of the cake's rim, because 90° is a quarter of the full 360°. Every arc works the same way: its share of the rim equals its angle's share of 360°.

Definition

An arc is a piece of the circle's boundary. Its central angle $\theta$ is the angle it makes at the centre.
Arc length $l = \frac{\theta}{360^\circ} \times 2\pi r$ .
Fractions worth knowing: $30^\circ$ is $\frac{1}{12}$ , $45^\circ$ is $\frac{1}{8}$ , $60^\circ$ is $\frac{1}{6}$ , $72^\circ$ is $\frac{1}{5}$ , $90^\circ$ is $\frac{1}{4}$ , $120^\circ$ is $\frac{1}{3}$ , $180^\circ$ is $\frac{1}{2}$ .
Working back: the whole circumference is $\frac{360^\circ}{\theta} \times l$ . Then $r = \frac{C}{2\pi}$ , and the circle's area $\pi r^2$ follows.
A chord equal to the radius makes an equilateral triangle with the two radii, so its arc has a $60^\circ$ angle.
The minor arc uses $\theta$ ; the major arc is the rest, $\frac{360^\circ - \theta}{360^\circ}$ of the circumference. Together they make the whole circle.
Fast route: find the fraction first, then take that part of $2\pi r$ . With $\theta = 40^\circ$ the arc is $\frac{1}{9}$ of the circumference, nothing more. Always safe.

Arc length

l = \frac{\theta}{360^\circ} \times 2\pi r \qquad C = \frac{360^\circ}{\theta} \times l

$\theta$ central angle in degrees
larc length

Worked example

An arc makes an angle of 40° at the centre of a circle of radius 63 cm. Find its length. (Take

\pi = \frac{22}{7}

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 26 Sep 2025, Shift 3 · Q56Easy

Example 1 · Mensuration 2D · Arc Length and Sector Perimeter

The angle subtended by an arc at the center is

72^\circ

, and its length is 22 cm. Find the radius of the circle.

Doubling a diameter that is already a diameter

If the stem gives the diameter, the circumference is

\pi d

. Writing

2\pi d

doubles the arc.

Taking the minor arc for the major one

A chord cuts the circle into two arcs. The smaller uses

\theta

; the larger uses

360^\circ - \theta

. Read which one the stem asks for.

Drill 9 more on arc length as a fraction of the circumference

Concept 2 of 3: Perimeter of a sector

A sector is a slice of a circle, like a slice of pizza. Walk round its edge and you cover the curved crust and the two straight cut edges. The cut edges are radii.

Definition

A sector is the region between two radii and the arc joining them.
Perimeter of a sector $=$ arc $+ 2r$ .
A semicircle's perimeter is $\pi r + 2r$ .
When a piece is cut away from a round object, the piece left over is also a sector: its angle is $360^\circ$ minus the cut angle, and its perimeter still includes both straight cut edges.
Fast route: with $\pi = \frac{22}{7}$ and a radius that is a multiple of 7, the circumference is a whole number, so find it first and take the fraction. Safe when the angle's fraction divides it evenly; otherwise keep the fraction to the end.

Perimeter of a sector

P = 2r + \frac{\theta}{360^\circ} \times 2\pi r \qquad P_{\text{semicircle}} = \pi r + 2r

Worked example

Find the perimeter of a sector of radius 42 cm and angle 30°. (Take

\pi = \frac{22}{7}

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 13 Sep 2024, Shift 1 · Q74Easy

Example 2 · Mensuration 2D · Arc Length and Sector Perimeter

One-quarter of a circular pizza of radius 21 cm was removed from the whole pizza. What is the perimeter (in cm) of the remaining pizza? (Use

\pi = 22/7

)

Leaving out the radii

The arc alone is not the perimeter of a sector. Add both straight edges,

2r

Counting the radius once

A sector has TWO straight edges, one along each radius. A semicircle's straight edge is the diameter, which is also

2r

Drill 6 more on perimeter of a sector

Concept 3 of 3: Radians: arc length equals radius times angle

Lay the radius along the rim of the circle. The angle at the centre it covers is one radian. An arc twice as long as the radius covers two radians, so in radians an arc is simply radius times angle.

Definition

One radian is the central angle whose arc equals the radius. A full turn is $2\pi$ radians, so $\pi$ radians $= 180^\circ$ , and one radian is about $57.3^\circ$ .
Converting: degrees $=$ radians $\times \frac{180^\circ}{\pi}$ ; radians $=$ degrees $\times \frac{\pi}{180^\circ}$ .
Common angles: $\frac{\pi}{6} = 30^\circ$ , $\frac{\pi}{4} = 45^\circ$ , $\frac{\pi}{3} = 60^\circ$ , $\frac{\pi}{2} = 90^\circ$ , $\frac{2\pi}{3} = 120^\circ$ , $\frac{3\pi}{4} = 135^\circ$ , $\pi = 180^\circ$ .
With the angle in radians: arc $l = r\theta$ , so $\theta = \frac{l}{r}$ and $r = \frac{l}{\theta}$ .
The straight line joining the arc's ends, the chord, is $2r\sin\frac{\theta}{2}$ , always a little shorter than the arc.
Fast route: an angle with $\pi$ in it: replace $\pi$ by $180^\circ$ to read it in degrees. An angle given as a plain number (2, 1.5) is in radians: use $l = r\theta$ directly, never the 360 fraction.

Radian measure

l = r\theta \qquad \pi \text{ radians} = 180^\circ \qquad \theta_{\text{rad}} = \theta^\circ \times \frac{\pi}{180^\circ}

$\theta$ central angle in radians in the first formula

Worked example

An arc 15 cm long is cut from a circle of radius 6 cm. Find the angle at the centre in radians and in degrees.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 12 Sep 2025, Shift 2 · Q69Easy

Example 3 · Mensuration 2D · Arc Length and Sector Perimeter

What is the central angle of a sector with an arc length of 10 cm in a circle of radius 5 cm?

Putting degrees into l = rθ

The rule

l = r\theta

needs

\theta

in radians. With

60^\circ

, first write

\frac{\pi}{3}

; otherwise the arc comes out 60 times the radius.

Taking the chord for the arc

2r\sin\frac{\theta}{2}

is the straight chord between the arc's ends. The curved arc is

r\theta

, which is always longer.

Drill 1 more on radians: arc length equals radius times angle

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Arc length as a fraction of the circumference
Arc length
$l = \frac{\theta}{360^\circ} \times 2\pi r \qquad C = \frac{360^\circ}{\theta} \times l$
Perimeter of a sector
Perimeter of a sector
$P = 2r + \frac{\theta}{360^\circ} \times 2\pi r \qquad P_{\text{semicircle}} = \pi r + 2r$
Radians: arc length equals radius times angle
Radian measure
$l = r\theta \qquad \pi \text{ radians} = 180^\circ \qquad \theta_{\text{rad}} = \theta^\circ \times \frac{\pi}{180^\circ}$

Watch out for (6)

Doubling a diameter that is already a diameter
→ Arc length as a fraction of the circumference
Taking the minor arc for the major one
→ Arc length as a fraction of the circumference
Leaving out the radii
→ Perimeter of a sector
Counting the radius once
→ Perimeter of a sector
Putting degrees into l = rθ
→ Radians: arc length equals radius times angle
Taking the chord for the arc
→ Radians: arc length equals radius times angle

Test yourself on Mensuration 2D

20 past SSC CGL questions from this chapter, timed at 12 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

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