SSC CGL Maths · Mensuration 2D

Circumference, Wheels and Rings

A circle's boundary is 2πr long and it encloses πr²; a wheel moves one circumference per turn, and a ring is the difference of two circles.

All Mensuration 2D notes

Why this matters

Twenty-five questions in the bank, twenty-two EASY and three MODERATE, nineteen of them from the 2025 shifts. Circumference and area are the base for the arc and sector pages that follow, and the ring questions all reduce to one difference of squares.

Concept 1 of 3: Circumference, wheel revolutions and fencing a circle

Roll a wheel through one full turn and it lays its whole rim on the road. So the distance per turn is the circumference, and counting turns is just dividing a distance by that one length.

Definition

Circumference $C = 2\pi r = \pi d$ . For every circle, $\frac{C}{d} = \pi \approx 3.14$ .
Use $\pi = \frac{22}{7}$ when the radius is a multiple of 7, and $3.14$ otherwise, unless the stem says which.
Revolutions of a wheel $=$ distance covered $\div$ circumference. Change both to the same unit first.
Fencing cost $=$ circumference $\times$ rate per metre; turned round, circumference $=$ cost $\div$ rate.
Part of a turn covers the same part of the circumference: a quarter turn covers $\frac{1}{4}C$ , and a minute hand moves half its circle in 30 minutes.
$C$ is proportional to $r$ : circumferences, radii and diameters are all in the same ratio.

Circumference and revolutions

C = 2\pi r = \pi d \qquad N = \frac{\text{distance}}{C}

r, dradius and diameter
Nnumber of revolutions

Worked example

A wheel of radius 35 cm rolls 1.1 km along a road. How many revolutions does it make? (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, 15 Sep 2025, Shift 3 · Q70Easy

Example 1 · Mensuration 2D · Circumference, Wheels and Rings

The cost of fencing a circular ground at ₹120/m is ₹3,768. Find the radius of the ground.

A whole-number ratio for circumference to diameter

Circumference divided by diameter is

\pi

for every circle, a little over 3. A ratio such as

3 : 5

would make the circumference shorter than the diameter, which no circle can have.

Dividing kilometres by centimetres

For revolutions, put the distance and the circumference in the same unit. A speed in km/h and a radius in cm must both become cm (and per minute, if revolutions per minute are asked).

Drill 8 more on circumference, wheel revolutions and fencing a circle

Concept 2 of 3: Area of a circle and how it scales with the radius

A circle is a shape that keeps its form as it grows, so the scaling rule from triangles applies: area goes as the square of the radius. Double the radius and the area is four times as large.

Definition

Area $A = \pi r^2$ . Part of a circle (a quarter left, $90\%$ painted) is that part of $\pi r^2$ .
Area ratio $= (\text{radius ratio})^2 = (\text{circumference ratio})^2$ .
A radius down $10\%$ makes the area $0.81$ of the old, $19\%$ less.
One circle equal to two: if its area equals the sum of circles of radii $r_1$ and $r_2$ , then $R^2 = r_1^2 + r_2^2$ . The $\pi$ cancels, so a Pythagorean triple gives $R$ at once (6, 8 give 10).
Same boundary, different shape: a wire of length $L$ bent into a circle encloses $\frac{L^2}{4\pi}$ ; bent into a square it encloses $\frac{L^2}{16}$ . The circle wins by the factor $\frac{4}{\pi} \approx 1.27$ .

Area of a circle

A = \pi r^2 \qquad \frac{A_2}{A_1} = \left(\frac{r_2}{r_1}\right)^2 \qquad R^2 = r_1^2 + r_2^2

Rradius of the circle whose area is the sum of the other two

Worked example

The radius of a circular plate is increased from 7 cm to 21 cm. Find the new area, and how many times the old area it is. (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, 16 Sep 2025, Shift 1 · Q68Easy

Example 2 · Mensuration 2D · Circumference, Wheels and Rings

Two circular pizzas have radii in the ratio 2:3. If the smaller pizza has an area of

100\pi\ \text{cm}^2

, what is the area of the larger pizza?

Using a circumference ratio as an area ratio

Circumferences in the ratio

3 : 5

mean radii

3 : 5

and areas

9 : 25

. Square the length ratio before using it for area.

Adding radii to add areas

A circle with the area of two circles of radii 8 and 15 has

R = \sqrt{64 + 225} = 17

, not

8 + 15 = 23

. Radii add only when circumferences are added.

Drill 6 more on area of a circle and how it scales with the radius

Concept 3 of 3: Area of a ring, a circular track or a path round a circle

A ring is a big circle with a small circle taken out of its middle. Its area is the big area minus the small one, and writing that difference as a product makes the arithmetic short.

Definition

Ring with outer radius $R$ and inner radius $r$ : area $= \pi(R^2 - r^2) = \pi(R - r)(R + r)$ . The width is $R - r$ .
Path outside a circle of radius $r$ : $R = r + w$ . Path inside the edge of a circle of radius $R$ : $r = R - w$ .
Fast route: $\pi \times$ width $\times$ (sum of radii). For $r = 20$ , $w = 2.5$ : $\pi \times 2.5 \times 42.5$ . Always safe; it is the same difference of squares.
Chord of the outer circle touching the inner one: the radius to the touching point is perpendicular to the chord and halves it, so $R^2 - r^2 = \left(\frac{c}{2}\right)^2$ and the ring's area is $\pi\left(\frac{c}{2}\right)^2$ , with no radii needed.
Ratios: with $R : r = 3 : 2$ , areas are $9 : 4$ , so ring : inner circle $= 5 : 4$ .

Area of a ring

A = \pi(R^2 - r^2) = \pi(R - r)(R + r) \qquad A = \pi\left(\frac{c}{2}\right)^2

R, router and inner radii
cchord of the outer circle that touches the inner circle

Worked example

A circular pond of radius 21 m has a path 3.5 m wide around it. Find the area of the path. (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, 12 Sep 2025, Shift 2 · Q66Easy

Example 3 · Mensuration 2D · Circumference, Wheels and Rings

A circular park having a radius of 14 m. If a 1.5 m wide path is built around it, what is the approximate area of the path?

Squaring the width

A path 2 m wide round a circle of radius 10 m has area

\pi(12^2 - 10^2) = 44\pi

, not

\pi \times 2^2

. The width is a difference of radii, not a radius.

Swapping inner and outer radii

The outer radius is always the larger. For a path outside, add the width to the circle's radius; for a path inside, subtract it.

Drill 8 more on area of a ring, a circular track or a path round a circle

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)

Circumference, wheel revolutions and fencing a circle
Circumference and revolutions
$C = 2\pi r = \pi d \qquad N = \frac{\text{distance}}{C}$
Area of a circle and how it scales with the radius
Area of a circle
$A = \pi r^2 \qquad \frac{A_2}{A_1} = \left(\frac{r_2}{r_1}\right)^2 \qquad R^2 = r_1^2 + r_2^2$
Area of a ring, a circular track or a path round a circle
Area of a ring
$A = \pi(R^2 - r^2) = \pi(R - r)(R + r) \qquad A = \pi\left(\frac{c}{2}\right)^2$

Watch out for (6)

A whole-number ratio for circumference to diameter
→ Circumference, wheel revolutions and fencing a circle
Dividing kilometres by centimetres
→ Circumference, wheel revolutions and fencing a circle
Using a circumference ratio as an area ratio
→ Area of a circle and how it scales with the radius
Adding radii to add areas
→ Area of a circle and how it scales with the radius
Squaring the width
→ Area of a ring, a circular track or a path round a circle
Swapping inner and outer radii
→ Area of a ring, a circular track or a path round a circle

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 25 questions