CDS Mathematics · Mensuration 2D

Circumference, Wheels & Rings

Circumference and area of a circle, revolutions of a wheel, how area scales with the radius, and the ring between two concentric circles.

All Mensuration 2D notes CDS Mathematics strategy

Why this matters

The easiest page of the chapter and one of the most dependable: wheel-revolution questions have appeared in almost every paper. Nearly all of it runs on π = 22/7, with radii chosen as multiples of 7 so that the arithmetic comes out whole.

Concept 1 of 3: Circumference and wheel revolutions

One turn of a wheel rolls it forward by exactly one circumference. So revolutions are distance divided by circumference; convert both to the same unit first.

Definition

Circumference $= 2\pi r = \pi d$ .
Revolutions $= \dfrac{\text{distance}}{\pi d}$ .
Two wheels covering the same distance: revolutions are inversely proportional to the diameters, $n_1 d_1 = n_2 d_2$ .
Semicircle perimeter $= \pi r + 2r = \dfrac{36r}{7}$ with $\pi = \tfrac{22}{7}$ .
"Circumference exceeds diameter by $k$ ": $\pi d - d = k$ , so $d = \dfrac{7k}{15}$ .
On a circular track the inner wheel runs the smaller radius, so it turns fewer times.

Wheel revolutions

n = \frac{\text{distance}}{2\pi r}

Worked example

A wheel of radius

21

cm rolls

2.64

km. How many revolutions does it make?

(\pi = \tfrac{22}{7})

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2016 · CDS (II) 2016 — Elementary Mathematics · Q67Moderate

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

The wheels of a car are of diameter 80 cm each. The car is travelling at a speed of 66 km/hour. What is the number of complete revolutions each wheel makes in 10 minutes ?

Radius or diameter?

Stems switch between the two. Revolutions use the circumference

\pi d = 2\pi r

; dividing by

\pi r

doubles the answer and that doubled value is usually an option.

Drill 8 more on circumference and wheel revolutions

Concept 2 of 3: Area grows as the square of the radius

Circumference is proportional to the radius and area to its square. A

10\%

longer circumference is a

10\%

larger radius and a

21\%

larger area.

Definition

Area $= \pi r^2$ .
Areas in ratio $p : q$ means radii in ratio $\sqrt p : \sqrt q$ .
Radius scaled by $k$ : circumference by $k$ , area by $k^2$ .
A circle equal in area to two others: $R^2 = r_1^2 + r_2^2$ .
Cutting holes from a plate of uniform thickness removes the same fraction of weight as of area.

Circle

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

Worked example

The radius of a circle is increased by

20\%

. By what percentage does its area increase?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2017 · CDS (I) 2017 — Elementary Mathematics · Q88Moderate

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

The radius of a circle is increased so that its circumference increases by 15%. The area of the circle will increase by

Equal-area rings get thinner outward

Split a disc into rings of equal area and the radii go as

\sqrt1, \sqrt2, \sqrt3, \ldots

. The ratio of neighbouring radii,

\sqrt{1 + \tfrac1m}

, falls as you move out.

Drill 5 more on area grows as the square of the radius

Concept 3 of 3: The ring between two circles

The area between two concentric circles is

\pi(R^2 - r^2)

, and

R^2 - r^2

is often all you know. A chord of the outer circle that just touches the inner one gives it directly: half the chord squared.

Definition

Ring (annulus) area $= \pi(R^2 - r^2) = \pi(R + r)(R - r)$ .
A path of width $w$ round a circular plot of radius $r$ : outer radius $r + w$ .
A chord of length $2c$ of the outer circle tangent to the inner circle: $R^2 - r^2 = c^2$ , so the ring's area is $\pi c^2$ , whatever the two radii are.

Ring from a tangent chord

\text{Ring} = \pi(R^2 - r^2) = \pi\left(\tfrac{\text{chord}}{2}\right)^2

Worked example

A chord of length

28

cm of the outer of two concentric circles touches the inner circle. Find the area of the ring.

(\pi = \tfrac{22}{7})

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2024 · CDS (I) 2024 — Elementary Mathematics · Q40Moderate

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

What is the area of the region between two concentric circles, if the length of a chord of the outer circle touching the inner circle at a particular point of its circumference is 14 cm ? (Take

\pi = \frac{22}{7}

)

"Cannot be determined" is the bait

The ring's area needs only

R^2 - r^2

, not

R

and

r

separately. When an option says the data are insufficient, the tangent-chord fact is usually what makes it wrong.

Drill 3 more on the ring between two circles

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 and wheel revolutions
Wheel revolutions
$n = \frac{\text{distance}}{2\pi r}$
Area grows as the square of the radius
Circle
$A = \pi r^2, \qquad \frac{A_1}{A_2} = \left(\frac{r_1}{r_2}\right)^2$
The ring between two circles
Ring from a tangent chord
$\text{Ring} = \pi(R^2 - r^2) = \pi\left(\tfrac{\text{chord}}{2}\right)^2$

Watch out for (3)

Radius or diameter?
→ Circumference and wheel revolutions
Equal-area rings get thinner outward
→ Area grows as the square of the radius
"Cannot be determined" is the bait
→ The ring between two circles

Test yourself on Mensuration 2D

20 past CDS questions from this chapter, timed at 24 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 19 questions