A Wheel's Circumference Is the Distance of One Full Turn
Find a circle's area and circumference either way round, compute the area of a ring, solve path-and-cost problems for rectangular and circular fields, and count the revolutions of a wheel.
How far does a wheel travel in one complete turn?
Exactly its own circumference. Roll a cycle wheel through one full turn and the mark you started from returns to the ground having laid the whole rim out along the road.
A cycle wheel of diameter cm therefore covers
in one revolution. That is just over three times the wheel's own height, which is the whole meaning of being a little more than : the way round a circle is a bit more than three times the way across it.
And that ratio is the same for every circle — a bangle, a tawa, a stadium track or the equator. Divide any circle's circumference by its diameter and you get the same number:
**Use when the radius is a multiple of ** and the arithmetic stays exact; use otherwise.
The one distinction to fix now. The circumference goes round the boundary and is measured in centimetres or metres. The area covers the inside and is measured in square centimetres or square metres. They use the same and the same radius, and mixing them is the most penalised error in this chapter.
This page covers the second part of the ICSE Class 9 Mathematics chapter on area and perimeter: the circle's area and circumference, the ring between two concentric circles, paths around fields with their costs, and the revolutions of a wheel.
A cycle wheel of diameter cm therefore covers
in one revolution. That is just over three times the wheel's own height, which is the whole meaning of being a little more than : the way round a circle is a bit more than three times the way across it.
And that ratio is the same for every circle — a bangle, a tawa, a stadium track or the equator. Divide any circle's circumference by its diameter and you get the same number:
**Use when the radius is a multiple of ** and the arithmetic stays exact; use otherwise.
The one distinction to fix now. The circumference goes round the boundary and is measured in centimetres or metres. The area covers the inside and is measured in square centimetres or square metres. They use the same and the same radius, and mixing them is the most penalised error in this chapter.
This page covers the second part of the ICSE Class 9 Mathematics chapter on area and perimeter: the circle's area and circumference, the ring between two concentric circles, paths around fields with their costs, and the revolutions of a wheel.
Formula
How do you find a circle's area and circumference, and work backwards?
Two formulas, both built on the radius.
Worked example 1 — forwards. Find the circumference and area of a circle of radius cm.
Worked example 2 — from the area. The area of a circle is cm. Find its radius and circumference.
Worked example 3 — from the circumference. A circle has circumference cm. Find its radius and area.
A relation worth knowing, because it checks every answer. Comparing the two formulas,
So the area of a circle is half its circumference times its radius — the same shape as for a triangle, which is no accident: cut the circle into very many thin sectors and each is almost a triangle of height , with all the bases adding up to .
Check example 1 with it: cm, as required.
The mistake this section exists to prevent. A question giving the diameter needs it halved before it goes into . A wheel of diameter cm has radius cm and area cm — using as the radius would give four times too much, because area depends on the square of the radius. Read the word diameter every time it appears and halve it on the first line.
Worked example 1 — forwards. Find the circumference and area of a circle of radius cm.
Worked example 2 — from the area. The area of a circle is cm. Find its radius and circumference.
Worked example 3 — from the circumference. A circle has circumference cm. Find its radius and area.
A relation worth knowing, because it checks every answer. Comparing the two formulas,
So the area of a circle is half its circumference times its radius — the same shape as for a triangle, which is no accident: cut the circle into very many thin sectors and each is almost a triangle of height , with all the bases adding up to .
Check example 1 with it: cm, as required.
The mistake this section exists to prevent. A question giving the diameter needs it halved before it goes into . A wheel of diameter cm has radius cm and area cm — using as the radius would give four times too much, because area depends on the square of the radius. Read the word diameter every time it appears and halve it on the first line.
How do you find the area of a ring between two circles?
Subtract the inner circle's area from the outer one's. Two concentric circles — same centre, different radii — enclose a ring, also called an annulus.
The factorised form is usually faster, since is the width of the ring.
Worked example 1. Two concentric circles have radii cm and cm. Find the area of the ring between them.
Or by the factorised form: cm, as required.
Worked example 2 — from the two circumferences. The outer and inner circumferences of a circular track are m and m. Find the width of the track and its area.
Notice the shortcut in that last line. Dividing by gave exactly, so the whole calculation stayed free of decimals until the end. Cancel before multiplying and problems rarely need a calculator.
A boundary case that tests understanding. If the two circles have the same centre and nearly the same radius, the ring becomes a thin strip. Its area is then close to , the circumference times the width — as if the strip had been cut open and straightened into a long rectangle.
Check that approximation: with m and m, the strip estimate gives m against the true m. The estimate is low because the outer edge is longer than the inner one, and the exact formula uses the average of the two circumferences — which is what quietly supplies.
The factorised form is usually faster, since is the width of the ring.
Worked example 1. Two concentric circles have radii cm and cm. Find the area of the ring between them.
Or by the factorised form: cm, as required.
Worked example 2 — from the two circumferences. The outer and inner circumferences of a circular track are m and m. Find the width of the track and its area.
Notice the shortcut in that last line. Dividing by gave exactly, so the whole calculation stayed free of decimals until the end. Cancel before multiplying and problems rarely need a calculator.
A boundary case that tests understanding. If the two circles have the same centre and nearly the same radius, the ring becomes a thin strip. Its area is then close to , the circumference times the width — as if the strip had been cut open and straightened into a long rectangle.
Check that approximation: with m and m, the strip estimate gives m against the true m. The estimate is low because the outer edge is longer than the inner one, and the exact formula uses the average of the two circumferences — which is what quietly supplies.
How do you solve a path-and-cost problem around a field?
Find the larger area and the smaller area separately, subtract to get the path, then multiply by the rate. Whether the path is inside or outside decides which rectangle is which.
Worked example 1 — a path outside a rectangular field. A field measures m by m, and a path m wide runs all round it on the outside. Find the area of the path and the cost of paving it at per square metre.
The path adds m on both sides of each dimension, so
**The rather than is where most marks are lost.** A path of width all round changes each dimension by , because it appears at both ends. Sketch the field with the path drawn and the doubling becomes obvious.
Worked example 2 — the same path inside. If instead the m path runs inside the boundary of the same field, find its area.
Inside and outside give different answers — m against m — because the outer path wraps round a longer boundary. Read which side the path is on before writing anything.
Worked example 3 — a circular park. A circular park of radius m has a path m wide around it. Find the area of the path and the cost of gravelling it at per square metre.
Here the width is added only once, because a radius runs outward from the centre in one direction only. That is the difference between the circular and the rectangular case, and it is the most common confusion in this section: a rectangle grows at both ends, a radius at one.
Worked example 4 — two crossing paths. A rectangular field is m by m. Two paths, each m wide, run through the middle — one parallel to the length and one parallel to the breadth. Find the total area of the paths.
But the square where they cross has been counted twice, so subtract it once:
Forgetting the overlap is the whole point of this question type. The crossing is m by m, it belongs to both paths, and it must be removed exactly once.
Worked example 1 — a path outside a rectangular field. A field measures m by m, and a path m wide runs all round it on the outside. Find the area of the path and the cost of paving it at per square metre.
The path adds m on both sides of each dimension, so
**The rather than is where most marks are lost.** A path of width all round changes each dimension by , because it appears at both ends. Sketch the field with the path drawn and the doubling becomes obvious.
Worked example 2 — the same path inside. If instead the m path runs inside the boundary of the same field, find its area.
Inside and outside give different answers — m against m — because the outer path wraps round a longer boundary. Read which side the path is on before writing anything.
Worked example 3 — a circular park. A circular park of radius m has a path m wide around it. Find the area of the path and the cost of gravelling it at per square metre.
Here the width is added only once, because a radius runs outward from the centre in one direction only. That is the difference between the circular and the rectangular case, and it is the most common confusion in this section: a rectangle grows at both ends, a radius at one.
Worked example 4 — two crossing paths. A rectangular field is m by m. Two paths, each m wide, run through the middle — one parallel to the length and one parallel to the breadth. Find the total area of the paths.
But the square where they cross has been counted twice, so subtract it once:
Forgetting the overlap is the whole point of this question type. The crossing is m by m, it belongs to both paths, and it must be removed exactly once.
How many turns does a wheel make over a given distance?
Divide the distance by the circumference — after converting both into the same unit.
Worked example 1. A wheel of diameter cm rolls along a road. How many revolutions does it make in covering km?
First the circumference:
Now the distance in the same unit: km m.
Worked example 2 — the other way round. A wheel of radius cm makes revolutions. What distance does it cover?
Worked example 3 — bringing in speed. A wheel of diameter cm turns times a minute. Find the speed in kilometres per hour.
Worked example 4 — backwards from a speed. How many revolutions per minute must the same wheel make to travel at km/h?
The unit conversion is where these questions are won or lost, not the division. Write the circumference in metres on the first line and convert the distance to metres on the second, and the rest is arithmetic.
And one comparison worth making. A larger wheel covers the same road in fewer turns. A cart wheel of diameter cm has circumference m, so it needs only
for that same km — exactly half as many as the cm wheel. Doubling the diameter halves the number of turns, because circumference is proportional to diameter. Note that this is a linear relationship, unlike area, which would have gone up four times.
Worked example 1. A wheel of diameter cm rolls along a road. How many revolutions does it make in covering km?
First the circumference:
Now the distance in the same unit: km m.
Worked example 2 — the other way round. A wheel of radius cm makes revolutions. What distance does it cover?
Worked example 3 — bringing in speed. A wheel of diameter cm turns times a minute. Find the speed in kilometres per hour.
Worked example 4 — backwards from a speed. How many revolutions per minute must the same wheel make to travel at km/h?
The unit conversion is where these questions are won or lost, not the division. Write the circumference in metres on the first line and convert the distance to metres on the second, and the rest is arithmetic.
And one comparison worth making. A larger wheel covers the same road in fewer turns. A cart wheel of diameter cm has circumference m, so it needs only
for that same km — exactly half as many as the cm wheel. Doubling the diameter halves the number of turns, because circumference is proportional to diameter. Note that this is a linear relationship, unlike area, which would have gone up four times.
Exam tip
What layout keeps circle calculations accurate?
Write down the radius as its own first line, especially when the question gives a diameter. Half the errors in this chapter are made before any formula is used.
- Convert diameter to radius immediately: * cm, so cm*. Then nothing downstream can go wrong
- **Choose deliberately.** Use when the radius or diameter is a multiple of ; use otherwise, and say which you used
- Cancel before multiplying. In , divide by first to get — much safer than multiplying by and then dividing
- For a path, sketch it and mark the width at both ends of a rectangle. Each dimension changes by ; a radius changes by only
- Subtract the overlap once when two paths cross
- For revolutions, put the circumference in metres and the distance in metres before dividing
- Multiply by the rate only at the very end, and write the currency symbol with the answer
- **Check with whenever you have both the area and the circumference
The misconception to name. Doubling the radius does not double the area — it quadruples** it, while the circumference only doubles. A circle of radius cm has area cm; one of radius cm has area cm, four times as much, while the circumferences are cm and cm. Lengths scale once, areas scale twice, and questions comparing two circles are usually testing exactly that.
- Convert diameter to radius immediately: * cm, so cm*. Then nothing downstream can go wrong
- **Choose deliberately.** Use when the radius or diameter is a multiple of ; use otherwise, and say which you used
- Cancel before multiplying. In , divide by first to get — much safer than multiplying by and then dividing
- For a path, sketch it and mark the width at both ends of a rectangle. Each dimension changes by ; a radius changes by only
- Subtract the overlap once when two paths cross
- For revolutions, put the circumference in metres and the distance in metres before dividing
- Multiply by the rate only at the very end, and write the currency symbol with the answer
- **Check with whenever you have both the area and the circumference
The misconception to name. Doubling the radius does not double the area — it quadruples** it, while the circumference only doubles. A circle of radius cm has area cm; one of radius cm has area cm, four times as much, while the circumferences are cm and cm. Lengths scale once, areas scale twice, and questions comparing two circles are usually testing exactly that.
Did you know
Why does a cycle's distance meter go wrong if you change the tyre?
A simple cycle meter does not measure distance at all. It counts revolutions of the wheel, using a magnet on a spoke, and then multiplies by a circumference stored in its memory. The distance shown is a calculation, not a measurement.
So the reading is only as good as that stored number. Suppose a meter is set for a wheel of diameter cm, whose circumference is
and the wheel is later replaced by one of diameter cm, with circumference
Now over turns the meter still reports m, while the cycle has actually travelled only m. **The meter over-reads by about m in every km**, which works out as
— and the same error appears in every speed it shows, since speed is that distance divided by time.
The same arithmetic runs in reverse for a heavy vehicle. A truck wheel of diameter m covers m per turn, so it turns half as often as a cycle wheel over the same road. That is why a large wheel rolls more easily over a pothole and why gear ratios are quoted along with wheel size: the road distance per turn is a property of the circumference and nothing else.
And it explains the oldest way of measuring a road. A wheel of known circumference pushed along the ground, with a counter on the axle, converts a long distance into a count of turns — the surveyor's wheel still used to measure footpaths and road widths. One circle's worth of arithmetic turns a counting problem into a distance, which is exactly what this chapter is for.
So the reading is only as good as that stored number. Suppose a meter is set for a wheel of diameter cm, whose circumference is
and the wheel is later replaced by one of diameter cm, with circumference
Now over turns the meter still reports m, while the cycle has actually travelled only m. **The meter over-reads by about m in every km**, which works out as
— and the same error appears in every speed it shows, since speed is that distance divided by time.
The same arithmetic runs in reverse for a heavy vehicle. A truck wheel of diameter m covers m per turn, so it turns half as often as a cycle wheel over the same road. That is why a large wheel rolls more easily over a pothole and why gear ratios are quoted along with wheel size: the road distance per turn is a property of the circumference and nothing else.
And it explains the oldest way of measuring a road. A wheel of known circumference pushed along the ground, with a counter on the axle, converts a long distance into a count of turns — the surveyor's wheel still used to measure footpaths and road widths. One circle's worth of arithmetic turns a counting problem into a distance, which is exactly what this chapter is for.
Exam relevance
Where do circle formulas reappear in JEE and NEET questions?
This is foundation work whose formulas are used far more often in Physics than in Mathematics papers.
Where it leads in Mathematics. Class 10 extends these to sectors and segments, where the area is a fraction of — the same idea as the arc in the circle chapter. In Class 11, angles move to radian measure, where and the full circle is radians; the circumference formula is that statement with . From there the circle becomes an equation in coordinate geometry, a JEE Main topic in its own right.
Where it leads in Physics. The revolutions calculation is the whole of rolling motion: a wheel rolling without slipping advances per turn, which is why holds for a rolling body. In Circular Motion, the period and speed are linked by — your circumference divided by the time for one turn. Both JEE Main and NEET set numericals on these, and the geometry is exactly what you have just done.
Worth noticing: the relation you checked above is the flat version of a result you meet again for the sphere and the cone, where a curved surface is built from many thin triangles or sectors. The thin-sector argument is the beginning of integration.
Question types to expect. At this level: area, circumference, rings, paths, costs and revolutions. In competitive papers: rolling-motion numericals, period and frequency of circular motion, and coordinate-geometry questions on circles. Assertion-reason items like the scaling question — double the radius and the area quadruples while the circumference doubles.
The single trap that costs marks. Using the diameter as the radius. In a Physics rolling question this appears as using in , and it makes every following number twice too large. **Write on its own line, always.
A second trap. Unit conversion. A radius in centimetres with a distance in kilometres needs both brought to metres before dividing, and a competitive paper will not warn you.
Board versus competitive emphasis. ICSE marks the formula, the substitution, the unit and the cost line; a competitive paper marks one number, usually inside a Physics question. The transferable sentence is one turn of a wheel advances it by one circumference** — that single fact answers a surprising share of rolling-motion questions.
Where it leads in Mathematics. Class 10 extends these to sectors and segments, where the area is a fraction of — the same idea as the arc in the circle chapter. In Class 11, angles move to radian measure, where and the full circle is radians; the circumference formula is that statement with . From there the circle becomes an equation in coordinate geometry, a JEE Main topic in its own right.
Where it leads in Physics. The revolutions calculation is the whole of rolling motion: a wheel rolling without slipping advances per turn, which is why holds for a rolling body. In Circular Motion, the period and speed are linked by — your circumference divided by the time for one turn. Both JEE Main and NEET set numericals on these, and the geometry is exactly what you have just done.
Worth noticing: the relation you checked above is the flat version of a result you meet again for the sphere and the cone, where a curved surface is built from many thin triangles or sectors. The thin-sector argument is the beginning of integration.
Question types to expect. At this level: area, circumference, rings, paths, costs and revolutions. In competitive papers: rolling-motion numericals, period and frequency of circular motion, and coordinate-geometry questions on circles. Assertion-reason items like the scaling question — double the radius and the area quadruples while the circumference doubles.
The single trap that costs marks. Using the diameter as the radius. In a Physics rolling question this appears as using in , and it makes every following number twice too large. **Write on its own line, always.
A second trap. Unit conversion. A radius in centimetres with a distance in kilometres needs both brought to metres before dividing, and a competitive paper will not warn you.
Board versus competitive emphasis. ICSE marks the formula, the substitution, the unit and the cost line; a competitive paper marks one number, usually inside a Physics question. The transferable sentence is one turn of a wheel advances it by one circumference** — that single fact answers a surprising share of rolling-motion questions.
Key takeaways
What should you be able to calculate about circles before moving on?
Two formulas, one subtraction and one division cover the whole of this chapter.
- ** and **, with when the radius is a multiple of
- Halve a given diameter on the first line, since area depends on
- Work backwards by rearranging: from a circumference, from an area
- ** is a free check whenever you have both
- Ring**: , where is the width
- **A path round a rectangle changes each dimension by **; a path round a circle changes the radius by only
- Crossing paths: add the two strips and subtract the overlap once
- Cost area rate, multiplied in at the very end
- Revolutions distance circumference, with both in the same unit
- Doubling the radius doubles the circumference and quadruples the area
The fastest self-test is the wheel. A wheel of diameter cm covering km — write down the number of revolutions in two lines, then work out what speed revolutions a minute represents in km/h.
- ** and **, with when the radius is a multiple of
- Halve a given diameter on the first line, since area depends on
- Work backwards by rearranging: from a circumference, from an area
- ** is a free check whenever you have both
- Ring**: , where is the width
- **A path round a rectangle changes each dimension by **; a path round a circle changes the radius by only
- Crossing paths: add the two strips and subtract the overlap once
- Cost area rate, multiplied in at the very end
- Revolutions distance circumference, with both in the same unit
- Doubling the radius doubles the circumference and quadruples the area
The fastest self-test is the wheel. A wheel of diameter cm covering km — write down the number of revolutions in two lines, then work out what speed revolutions a minute represents in km/h.