SSC CGL Maths · Mensuration 2D

Sector Area

A sector's area is the same fraction of πr² as its angle is of 360°, or half its arc times its radius.

Why this matters

Thirty-two questions in the bank, thirty EASY and two MODERATE, fourteen of them from the 2025 shifts: the largest single page in this chapter. Three routes reach the same area (the angle fraction, half arc times radius, half r squared times the angle in radians), and the fast mark comes from picking the one that matches what the stem gives.

Concept 1 of 4: Sector area as a fraction of the circle's area

A sector takes the same share of the circle's area as its angle takes of 360°, exactly as the arc took that share of the circumference. A 90° sector is a quarter of the disc.

Definition

Sector area $A = \frac{\theta}{360^\circ} \times \pi r^2$ .
Working back for the angle: $\frac{\theta}{360^\circ} = \frac{A}{\pi r^2}$ . Divide the sector by the whole circle, then multiply by $360^\circ$ .
Working back for the radius: $r^2 = \frac{360^\circ}{\theta} \times \frac{A}{\pi}$ , then take the square root.
A circle cut into $n$ equal sectors: each one is $\frac{\pi r^2}{n}$ .
Fast route: find the circle's area once, then take the fraction. With $r = 21$ and $\pi = \frac{22}{7}$ , the circle is 1386, and a $60^\circ$ sector is $\frac{1386}{6} = 231$ . Always safe.

Area of a sector from its angle

A = \frac{\theta}{360^\circ} \times \pi r^2

$\theta$ central angle in degrees

Worked example

Find the area of a sector of radius 21 cm and angle 80°. (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 1 · Q57Easy

Example 1 · Mensuration 2D · Sector Area

A sector of radius 14 cm has an area of

154\text{ cm}^2

. Find the angle of the sector.

Using 2πr for an area

2\pi r

is the circumference, a length. A sector's AREA takes its fraction of

\pi r^2

Stopping at r squared

Working back from an area gives

r^2

. Take the square root before choosing an option:

r^2 = 225

means

r = 15

Drill 9 more on sector area as a fraction of the circle's area

Concept 2 of 4: Sector area from its arc: half the arc times the radius

Cut a sector into many thin slices and each one is nearly a triangle with its tip at the centre, its height the radius and its base a tiny bit of the arc. Add them up and the sector is a triangle with base equal to the arc and height equal to the radius.

Definition

Sector area $= \frac{1}{2} \times$ arc $\times$ radius. No angle is needed.
With the angle $\theta$ in radians, the arc is $r\theta$ , so the area is $\frac{1}{2}r^2\theta$ .
Working back for the arc: $l = \frac{2A}{r}$ .
An arc given as a multiple of the radius ( $l = kr$ ) gives area $\frac{1}{2}kr^2$ .
Fast route: when the stem gives the arc and the radius, use $\frac{1}{2}lr$ and never find the angle. When it gives an angle in radians, use $\frac{1}{2}r^2\theta$ . Both are safe; the angle must be in radians in the second.

Area of a sector from its arc

A = \frac{1}{2}lr \qquad A = \frac{1}{2}r^2\theta

larc length
$\theta$ central angle in radians

Worked example

A sector of radius 9 cm has an arc 10 cm long. Find its area and its angle in radians.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 23 Sep 2025, Shift 3 · Q69Easy

Example 2 · Mensuration 2D · Sector Area

The angle of a sector is

\frac{\pi}{4}

radians, and the radius of the circle is 8 cm. What is the area of the sector?

Putting degrees into half r squared theta

\frac{1}{2}r^2\theta

needs

\theta

in radians. For a

45^\circ

sector use

\frac{\pi}{4}

, or use the 360 fraction instead.

Dropping the half

Arc times radius is TWICE the sector's area, just as base times height is twice a triangle's area.

Drill 11 more on sector area from its arc: half the arc times the radius

Concept 3 of 4: Comparing two sectors given in degrees and in radians

Two sectors can only be compared once their angles are in the same unit. After that, the area depends on just two things, the angle and the square of the radius.

Definition

Change both angles to degrees (or both to radians) first. $\frac{\pi}{3}$ and $60^\circ$ are the same angle.
Same circle, same angle: equal areas, ratio $1 : 1$ .
Same radius: areas are in the ratio of the angles.
Same angle: areas are in the ratio of the squares of the radii.
In general: $A_1 : A_2 = \theta_1 r_1^2 : \theta_2 r_2^2$ .
When it breaks: comparing by angle alone is safe only when the stem says the radii are equal (or the sectors are in the same circle). Without that, the ratio is not fixed.

Ratio of two sector areas

\frac{A_1}{A_2} = \frac{\theta_1 r_1^2}{\theta_2 r_2^2}

$\theta_1, \theta_2$ the two angles, in the same unit

Worked example

Sector P has radius 6 cm and angle 100°. Sector Q has radius 9 cm and angle

\frac{\pi}{3}

radians. Find the ratio of their areas.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 17 Sep 2025, Shift 3 · Q67Easy

Example 3 · Mensuration 2D · Sector Area

A sector has a central angle of

180^\circ

and a radius of 12 cm. Another sector has a central angle of

\frac{\pi}{2}

radians and the same radius. What is the ratio of the area of the first sector to the area of the second sector?

Comparing a degree number with a radian number

60

and

\frac{\pi}{3} \approx 1.05

look very different but are the same angle. Convert before comparing, or a

1 : 1

ratio looks like

57 : 1

Comparing by angle when the radii are unknown

Equal angles give equal areas only in circles of equal radius. If the stem does not say the radii are equal, the angles alone do not fix the ratio.

Drill 7 more on comparing two sectors given in degrees and in radians

Concept 4 of 4: Finding a sector from its perimeter

The perimeter ties the arc to the radius: arc plus two radii. Give one more fact (the angle, or the area) and the radius is pinned down.

Definition

Perimeter $P = 2r + l$ .
Angle known: the arc is a fixed multiple of $r$ (for $90^\circ$ with $\pi = \frac{22}{7}$ , $l = \frac{11}{7}r$ ), so $P = r \times (2 + \text{that multiple})$ . Solve for $r$ .
Area known: $l = P - 2r$ and $\frac{1}{2}lr = A$ give $r(P - 2r) = 2A$ , a quadratic in $r$ . It has two roots; keep the one the stem allows (a whole number, or one that matches an option).
Fast route: put each option's radius into $r(P - 2r) = 2A$ and keep the one that fits. Safe in a multiple-choice question; it does not tell you the second root.

Sector perimeter and area together

P = 2r + l \qquad r(P - 2r) = 2A

Worked example

A sector of angle 90° has a perimeter of 50 cm. Find its radius. (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, 10 Sep 2024, Shift 3 · Q67Moderate

Example 4 · Mensuration 2D · Sector Area

What is the central angle of a sector of a circle whose area and perimeter are, respectively, equal to 209

\text{cm}^2

and 63 cm if it is given that its radius is a non-negative integer (roundup to one decimal place)?

Keeping the wrong root

The perimeter-and-area equation is a quadratic with two positive roots. Both make a real sector; the stem's condition (a whole number, or the options) picks one.

Drill 1 more on finding a sector from its perimeter

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 (4)

Sector area as a fraction of the circle's area
Area of a sector from its angle
$A = \frac{\theta}{360^\circ} \times \pi r^2$
Sector area from its arc: half the arc times the radius
Area of a sector from its arc
$A = \frac{1}{2}lr \qquad A = \frac{1}{2}r^2\theta$
Comparing two sectors given in degrees and in radians
Ratio of two sector areas
$\frac{A_1}{A_2} = \frac{\theta_1 r_1^2}{\theta_2 r_2^2}$
Finding a sector from its perimeter
Sector perimeter and area together
$P = 2r + l \qquad r(P - 2r) = 2A$

Watch out for (7)

Using 2πr for an area
→ Sector area as a fraction of the circle's area
Stopping at r squared
→ Sector area as a fraction of the circle's area
Putting degrees into half r squared theta
→ Sector area from its arc: half the arc times the radius
Dropping the half
→ Sector area from its arc: half the arc times the radius
Comparing a degree number with a radian number
→ Comparing two sectors given in degrees and in radians
Comparing by angle when the radii are unknown
→ Comparing two sectors given in degrees and in radians
Keeping the wrong root
→ Finding a sector from its perimeter

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.

Take the Mensuration 2D chapter test Drill all 32 questions