Roll the Same Sheet the Other Way and You Lose Half the Volume
Learn the curved and total surface area of a cylinder, find the volume of solid and hollow cylinders, work backwards to a radius or height, and solve melting and recasting problems.
Does it matter which way you roll a rectangular sheet into a tube?
Enormously — the two tubes hold quite different amounts.
Take a rectangular sheet cm by cm. Roll it so the cm edge becomes the circumference. Then , giving cm and a height of cm, so
Now roll the same sheet the other way, so the cm edge is the circumference. Then cm with a height of cm, and the volume works out to .
Same sheet, same curved surface area, and yet one tube holds more than twice as much as the other. The reason is that volume depends on while the surface depends on — so making the tube fatter pays off far more than making it taller. This page covers the second part of the ICSE Class 8 Mathematics chapter on surface area, volume and capacity.
Take a rectangular sheet cm by cm. Roll it so the cm edge becomes the circumference. Then , giving cm and a height of cm, so
Now roll the same sheet the other way, so the cm edge is the circumference. Then cm with a height of cm, and the volume works out to .
Same sheet, same curved surface area, and yet one tube holds more than twice as much as the other. The reason is that volume depends on while the surface depends on — so making the tube fatter pays off far more than making it taller. This page covers the second part of the ICSE Class 8 Mathematics chapter on surface area, volume and capacity.
Formula
What are the surface area and volume formulas for a cylinder?
For a right circular cylinder of radius and height :
Unless a question says otherwise, take .
Where the curved surface comes from. Unroll the curved surface and it becomes a rectangle. Its height is and its width is the circumference , so its area is . The total surface adds the two circular ends, each of area .
Choose radii that are multiples of seven. With , the seven cancels and the arithmetic stays exact — which is why examination questions use radii like , and cm.
Worked example 1. A solid cylinder has radius cm and height cm.
Check the two surface areas. The two ends are , and . They agree.
Worked example 2. A cylinder has radius cm and height cm.
An open cylinder has only one end. A cylindrical bucket or pipe open at the top has curved surface plus one circle:
For cm and cm that is — less than the closed figure of by exactly one circle of . Read whether the solid is closed, open at one end, or open at both, because each answer differs by a whole circle.
Unless a question says otherwise, take .
Where the curved surface comes from. Unroll the curved surface and it becomes a rectangle. Its height is and its width is the circumference , so its area is . The total surface adds the two circular ends, each of area .
Choose radii that are multiples of seven. With , the seven cancels and the arithmetic stays exact — which is why examination questions use radii like , and cm.
Worked example 1. A solid cylinder has radius cm and height cm.
Check the two surface areas. The two ends are , and . They agree.
Worked example 2. A cylinder has radius cm and height cm.
An open cylinder has only one end. A cylindrical bucket or pipe open at the top has curved surface plus one circle:
For cm and cm that is — less than the closed figure of by exactly one circle of . Read whether the solid is closed, open at one end, or open at both, because each answer differs by a whole circle.
How do you find the volume and capacity of a solid or hollow cylinder?
**For a solid cylinder use . For a hollow one, subtract the inner cylinder from the outer.**
where is the outer radius and the inner.
Worked example 1 — capacity of a tank. A cylindrical water tank has radius m and height m. Find its capacity in litres.
Since litres:
Worked example 2 — a well. A well of diameter m is dug m deep. What volume of earth is removed?
The radius is m, so
Halve the diameter first. Using m as the radius would give four times the correct answer, and it is the single commonest error in cylinder questions — the diameter is quoted far more often than the radius in real descriptions of pipes and wells.
Worked example 3 — a hollow cylinder. A metal pipe has outer radius cm, inner radius cm and length cm. Find the volume of metal.
Subtract the squares, not the radii. Writing gives , which is wrong. The identity shows why: the correct factor is , not .
Worked example 4 — a thinner pipe. Outer radius cm, inner radius cm, height cm.
Worked example 5 — capacity against volume of material. For the pipe in example 3, the metal occupies , but the water it can carry is the inner cylinder:
Volume of material and capacity are different questions about the same pipe. One uses and the other uses alone, and deciding which the question wants is the substance of the problem rather than the arithmetic.
where is the outer radius and the inner.
Worked example 1 — capacity of a tank. A cylindrical water tank has radius m and height m. Find its capacity in litres.
Since litres:
Worked example 2 — a well. A well of diameter m is dug m deep. What volume of earth is removed?
The radius is m, so
Halve the diameter first. Using m as the radius would give four times the correct answer, and it is the single commonest error in cylinder questions — the diameter is quoted far more often than the radius in real descriptions of pipes and wells.
Worked example 3 — a hollow cylinder. A metal pipe has outer radius cm, inner radius cm and length cm. Find the volume of metal.
Subtract the squares, not the radii. Writing gives , which is wrong. The identity shows why: the correct factor is , not .
Worked example 4 — a thinner pipe. Outer radius cm, inner radius cm, height cm.
Worked example 5 — capacity against volume of material. For the pipe in example 3, the metal occupies , but the water it can carry is the inner cylinder:
Volume of material and capacity are different questions about the same pipe. One uses and the other uses alone, and deciding which the question wants is the substance of the problem rather than the arithmetic.
How do you find the radius or height when the area or volume is given?
Substitute what you know and solve for the unknown, undoing the formula one factor at a time.
Worked example 1 — height from the curved surface area. A cylinder of radius cm has curved surface area .
Worked example 2 — height from the volume. A cylinder of radius cm has volume .
Worked example 3 — radius from the volume. A cylinder of height cm has volume .
Take the square root last. The equation gives , and a student who stops there answers cm instead of cm. Writing on its own line, then cm on the next, makes the missing step impossible to skip.
Worked example 4 — radius from the curved surface area. A cylinder of height cm has curved surface area .
Note that here appears to the first power, so no square root is needed. Volume questions need a root and curved-surface questions do not, and knowing which you are in saves a wasted step.
Worked example 5 — both dimensions from a ratio. The radius and height of a cylinder are in the ratio , and its volume is . Find both.
Write and :
So , giving cm and cm.
Check exactly, before rounding. Since and :
The volume comes back exactly, which confirms the working even though and themselves had to be rounded.
An awkward cube root is a hint, not an error. Ratio questions in this chapter usually give a clean ; when they do not, the answer is simply not a whole number, and rounding sensibly with the units stated is the correct response.
Worked example 1 — height from the curved surface area. A cylinder of radius cm has curved surface area .
Worked example 2 — height from the volume. A cylinder of radius cm has volume .
Worked example 3 — radius from the volume. A cylinder of height cm has volume .
Take the square root last. The equation gives , and a student who stops there answers cm instead of cm. Writing on its own line, then cm on the next, makes the missing step impossible to skip.
Worked example 4 — radius from the curved surface area. A cylinder of height cm has curved surface area .
Note that here appears to the first power, so no square root is needed. Volume questions need a root and curved-surface questions do not, and knowing which you are in saves a wasted step.
Worked example 5 — both dimensions from a ratio. The radius and height of a cylinder are in the ratio , and its volume is . Find both.
Write and :
So , giving cm and cm.
Check exactly, before rounding. Since and :
The volume comes back exactly, which confirms the working even though and themselves had to be rounded.
An awkward cube root is a hint, not an error. Ratio questions in this chapter usually give a clean ; when they do not, the answer is simply not a whole number, and rounding sensibly with the units stated is the correct response.
How do you solve melting and recasting problems?
The volume stays the same. Melting changes the shape and nothing else, so set the two volumes equal.
Worked example 1 — cuboid into a cylinder. A metal cuboid cm by cm by cm is melted and recast into a cylinder of radius cm. Find its height.
Worked example 2 — one cylinder into several. A cylinder of radius cm and height cm is melted and recast into smaller cylinders of radius cm and height cm. How many?
Halving the radius quarters the volume, which is why four small cylinders come from one large — and it is worth seeing that the answer could have been predicted from without computing either volume.
Worked example 3 — comparing two cylinders. Cylinder A has radius cm and height cm; cylinder B has radius cm and height cm. Which holds more, and by how much?
B holds exactly twice as much, although the two use the same two numbers swapped over. The ratio can be got directly:
Doubling the radius has four times the effect of doubling the height, because volume depends on . This is the same fact that made the rolled sheet in the opening section hold twice as much one way round as the other.
Worked example 4 — the rolled sheet, worked out. A sheet cm by cm rolled along the cm edge gives , so
Rolled the other way, gives cm with cm:
Surface area is unchanged and volume is not. Both tubes have the same curved surface of , since that is just the sheet. It is a good reminder that equal surface areas do not mean equal volumes, and the reverse holds too — which is why every question of this type must be answered by computing the quantity actually asked for.
Worked example 1 — cuboid into a cylinder. A metal cuboid cm by cm by cm is melted and recast into a cylinder of radius cm. Find its height.
Worked example 2 — one cylinder into several. A cylinder of radius cm and height cm is melted and recast into smaller cylinders of radius cm and height cm. How many?
Halving the radius quarters the volume, which is why four small cylinders come from one large — and it is worth seeing that the answer could have been predicted from without computing either volume.
Worked example 3 — comparing two cylinders. Cylinder A has radius cm and height cm; cylinder B has radius cm and height cm. Which holds more, and by how much?
B holds exactly twice as much, although the two use the same two numbers swapped over. The ratio can be got directly:
Doubling the radius has four times the effect of doubling the height, because volume depends on . This is the same fact that made the rolled sheet in the opening section hold twice as much one way round as the other.
Worked example 4 — the rolled sheet, worked out. A sheet cm by cm rolled along the cm edge gives , so
Rolled the other way, gives cm with cm:
Surface area is unchanged and volume is not. Both tubes have the same curved surface of , since that is just the sheet. It is a good reminder that equal surface areas do not mean equal volumes, and the reverse holds too — which is why every question of this type must be answered by computing the quantity actually asked for.
Exam tip
Exam tip: halve the diameter, and take the root last
Halve the diameter before substituting. A well of diameter m has radius m — using quadruples the answer.
Use and work with radii that are multiples of so the seven cancels exactly.
, , . Check that CSA plus two circles equals TSA.
Read whether the cylinder is closed or open — an open one has one circle instead of two, so , gives against .
For a hollow cylinder, use and subtract the squares: , not .
Material against capacity: material uses , capacity uses alone.
When working backwards from a volume, write on one line and cm on the next — the square root is the step most often skipped. From a curved surface area, is to the first power and needs no root.
For melting and recasting, set the two volumes equal; nothing else is conserved.
Remember that volume depends on , so doubling the radius beats doubling the height four to one.
And convert with litres, litre.
Use and work with radii that are multiples of so the seven cancels exactly.
, , . Check that CSA plus two circles equals TSA.
Read whether the cylinder is closed or open — an open one has one circle instead of two, so , gives against .
For a hollow cylinder, use and subtract the squares: , not .
Material against capacity: material uses , capacity uses alone.
When working backwards from a volume, write on one line and cm on the next — the square root is the step most often skipped. From a curved surface area, is to the first power and needs no root.
For melting and recasting, set the two volumes equal; nothing else is conserved.
Remember that volume depends on , so doubling the radius beats doubling the height four to one.
And convert with litres, litre.
Did you know
Why tins are the shape they are
A tin of food is a cylinder, and its proportions are not arbitrary. Fix the volume and ask which cylinder uses the least metal, and the arithmetic picks out a particular shape.
Try it with a volume of about . A tall thin tin of radius cm needs a height of about cm, giving a total surface of roughly . A short wide one of radius cm needs a height of about cm, giving about . In between, radius cm with height cm gives — less metal than either extreme.
The pattern is that very tall and very flat are both wasteful, and something in the middle is best. A tall tin spends too much metal on its long curved wall; a flat one spends too much on its two large circles. The cheapest shape balances the two, and it turns out to be one where the height is close to the diameter.
Which is roughly what a tin of food looks like. Not exactly — a manufacturer also has to consider shelves, printing, stacking and how a hand grips it — but the mathematics sets the target the design departs from.
The same reasoning runs the other way for a water tank, where the goal is the most capacity for a given amount of sheet. Then the answer favours a wide tank, because volume rewards radius twice over while the wall's area rewards it only once. It is the same against comparison that made one rolled sheet hold twice as much as the other.
Try it with a volume of about . A tall thin tin of radius cm needs a height of about cm, giving a total surface of roughly . A short wide one of radius cm needs a height of about cm, giving about . In between, radius cm with height cm gives — less metal than either extreme.
The pattern is that very tall and very flat are both wasteful, and something in the middle is best. A tall tin spends too much metal on its long curved wall; a flat one spends too much on its two large circles. The cheapest shape balances the two, and it turns out to be one where the height is close to the diameter.
Which is roughly what a tin of food looks like. Not exactly — a manufacturer also has to consider shelves, printing, stacking and how a hand grips it — but the mathematics sets the target the design departs from.
The same reasoning runs the other way for a water tank, where the goal is the most capacity for a given amount of sheet. Then the answer favours a wide tank, because volume rewards radius twice over while the wall's area rewards it only once. It is the same against comparison that made one rolled sheet hold twice as much as the other.
Key takeaways
Cylinders: quick revision
- , , , with unless told otherwise.
- The curved surface unrolls into a rectangle of height and width .
- cm, cm gives , , — and the two ends of account for the difference.
- cm, cm gives and .
- Open at one end: for , — one circle less.
- Halve the diameter first. A well of diameter m has radius m, so m deep gives .
- A tank of radius m and height m holds litres.
- Hollow cylinder: . With , , : . Subtract the squares — , not .
- , , gives of material, while its capacity is the inner cylinder alone.
- Backwards: with gives cm; with gives cm.
- with gives , so cm — take the root last. From a CSA, is to the first power.
- Ratio question: with gives , so cm and cm.
- Melting keeps the volume. A cuboid cm recast at radius cm gives cm.
- A cylinder , gives four cylinders of , , predictable from .
- Swapping and changes the volume: gives while gives — exactly twice.
- A cm sheet rolled one way gives and the other way , with the same curved surface of .
- Equal surface areas do not mean equal volumes.
Roll a sheet of paper both ways, fill each tube with rice and pour one into the other — watching the fatter tube win is the fastest way to believe that volume depends on the square of the radius.
- The curved surface unrolls into a rectangle of height and width .
- cm, cm gives , , — and the two ends of account for the difference.
- cm, cm gives and .
- Open at one end: for , — one circle less.
- Halve the diameter first. A well of diameter m has radius m, so m deep gives .
- A tank of radius m and height m holds litres.
- Hollow cylinder: . With , , : . Subtract the squares — , not .
- , , gives of material, while its capacity is the inner cylinder alone.
- Backwards: with gives cm; with gives cm.
- with gives , so cm — take the root last. From a CSA, is to the first power.
- Ratio question: with gives , so cm and cm.
- Melting keeps the volume. A cuboid cm recast at radius cm gives cm.
- A cylinder , gives four cylinders of , , predictable from .
- Swapping and changes the volume: gives while gives — exactly twice.
- A cm sheet rolled one way gives and the other way , with the same curved surface of .
- Equal surface areas do not mean equal volumes.
Roll a sheet of paper both ways, fill each tube with rice and pour one into the other — watching the fatter tube win is the fastest way to believe that volume depends on the square of the radius.