Volume Is Just a Cross-Section Dragged Along a Length
Find the volume and surface areas of cubes and cuboids, handle open boxes and painting costs, work out the material in a hollow box, and use cross-section times length for beams, pipes and canals.
What is the one idea behind every volume formula?
Take a flat shape, and drag it straight along a direction at right angles to itself. The space it sweeps out is a solid, and its volume is
A brick is a rectangle dragged sideways. A pipe is a ring dragged along its axis. A canal is a rectangular channel dragged for kilometres. All three use the same sentence, and in Class 10 a cylinder will use it too, with a circle as the cross-section.
For a cuboid this gives the familiar formula straight away. The cross-section is a rectangle of area , dragged through a height , so
Worked example. A brick measures cm by cm by cm. Its cross-section is cm, dragged through cm, so
Any face can be the cross-section. Taking the by face instead gives cm, the same answer — which is the check to run whenever a question names a face as the cross-section.
And the units tell you which quantity you are computing. Length in cm, area in cm, volume in cm — one length, two lengths, three lengths. **A volume answer in cm is wrong before the number is even read.**
This page covers the ICSE Class 9 Mathematics chapter on solids: volume and surface areas of cubes and cuboids, open boxes and costs, hollow solids, and the cross-section rule.
A brick is a rectangle dragged sideways. A pipe is a ring dragged along its axis. A canal is a rectangular channel dragged for kilometres. All three use the same sentence, and in Class 10 a cylinder will use it too, with a circle as the cross-section.
For a cuboid this gives the familiar formula straight away. The cross-section is a rectangle of area , dragged through a height , so
Worked example. A brick measures cm by cm by cm. Its cross-section is cm, dragged through cm, so
Any face can be the cross-section. Taking the by face instead gives cm, the same answer — which is the check to run whenever a question names a face as the cross-section.
And the units tell you which quantity you are computing. Length in cm, area in cm, volume in cm — one length, two lengths, three lengths. **A volume answer in cm is wrong before the number is even read.**
This page covers the ICSE Class 9 Mathematics chapter on solids: volume and surface areas of cubes and cuboids, open boxes and costs, hollow solids, and the cross-section rule.
Formula
What are the volume and surface area formulas for a cube and a cuboid?
Volume multiplies the three dimensions; total surface area adds the six faces in three matching pairs.
For a cuboid of length , breadth and height :
For a cube of edge , all three dimensions are equal:
Lateral surface area means the four walls only — the sides you would paint on a room, leaving out the floor and the ceiling.
Worked example 1 — a full cuboid. A cuboid measures cm by cm by cm. Find its volume, total surface area, lateral surface area and diagonal.
Check the two surface areas against each other: the difference between total and lateral must be the top and bottom, that is cm. And cm, as required.
Worked example 2 — a cube. For a cube of edge cm,
Worked example 3 — working backwards. The volume of a cube is cm. Find its edge, surface area and diagonal.
Notice that the volume and the surface area came out as the same number here, , in different units. That is a coincidence of the edge being , and it is a useful reminder: **cm and cm are different quantities, and equal numbers mean nothing.
Where the diagonal formula comes from.** Apply the Pythagoras theorem twice: first across the base to get , then up the height, using that as one leg. The three-dimensional diagonal is two two-dimensional ones, and the derivation is worth doing once so the formula is never misremembered.
For a cuboid of length , breadth and height :
For a cube of edge , all three dimensions are equal:
Lateral surface area means the four walls only — the sides you would paint on a room, leaving out the floor and the ceiling.
Worked example 1 — a full cuboid. A cuboid measures cm by cm by cm. Find its volume, total surface area, lateral surface area and diagonal.
Check the two surface areas against each other: the difference between total and lateral must be the top and bottom, that is cm. And cm, as required.
Worked example 2 — a cube. For a cube of edge cm,
Worked example 3 — working backwards. The volume of a cube is cm. Find its edge, surface area and diagonal.
Notice that the volume and the surface area came out as the same number here, , in different units. That is a coincidence of the edge being , and it is a useful reminder: **cm and cm are different quantities, and equal numbers mean nothing.
Where the diagonal formula comes from.** Apply the Pythagoras theorem twice: first across the base to get , then up the height, using that as one leg. The three-dimensional diagonal is two two-dimensional ones, and the derivation is worth doing once so the formula is never misremembered.
How do you handle an open box and a cost-of-covering question?
Count the faces the question actually asks for — an open box has five, walls alone are four, and a room with a ceiling has five. Then multiply the area by the rate, at the very end.
Worked example 1 — an open box. A box without a lid measures cm by cm by cm. Find the area of card needed.
Five faces: the base plus four walls.
**Compare with the closed box's cm**: the difference is exactly one by face, cm. Subtracting the missing face from the total is usually faster than adding five faces one by one.
Worked example 2 — a room. A room is m long, m wide and m high. Find the cost of white-washing its four walls and ceiling at per square metre.
The floor is not white-washed, which is why it is left out. A question that says four walls and ceiling means five faces; one that says four walls means four. Underline the faces named in the question before you write a formula.
Worked example 3 — a different rate for a different surface. For the same room, find the cost of tiling the floor at per square metre.
Worked example 4 — a cube-shaped tank. A closed cubical tank has edge m. Find the area to be painted and the cost at per square metre.
Worked example 5 — doors and windows. If the room in example 2 has a door of m by m and two windows each m by m, find the wall area to be painted.
Openings are subtracted from the walls, not from the ceiling. That sounds obvious and is regularly got wrong when the arithmetic is done in a single long line. Compute the wall area, then subtract the openings, then apply the rate — three separate steps, three chances to be right.
Worked example 1 — an open box. A box without a lid measures cm by cm by cm. Find the area of card needed.
Five faces: the base plus four walls.
**Compare with the closed box's cm**: the difference is exactly one by face, cm. Subtracting the missing face from the total is usually faster than adding five faces one by one.
Worked example 2 — a room. A room is m long, m wide and m high. Find the cost of white-washing its four walls and ceiling at per square metre.
The floor is not white-washed, which is why it is left out. A question that says four walls and ceiling means five faces; one that says four walls means four. Underline the faces named in the question before you write a formula.
Worked example 3 — a different rate for a different surface. For the same room, find the cost of tiling the floor at per square metre.
Worked example 4 — a cube-shaped tank. A closed cubical tank has edge m. Find the area to be painted and the cost at per square metre.
Worked example 5 — doors and windows. If the room in example 2 has a door of m by m and two windows each m by m, find the wall area to be painted.
Openings are subtracted from the walls, not from the ceiling. That sounds obvious and is regularly got wrong when the arithmetic is done in a single long line. Compute the wall area, then subtract the openings, then apply the rate — three separate steps, three chances to be right.
How do you find the volume of material in a hollow box?
Subtract the inside volume from the outside volume — and reduce each dimension by twice the thickness, unless that dimension has an open end.
Worked example 1 — a closed hollow box. A closed box has external dimensions cm by cm by cm and its walls are cm thick. Find the volume of material used and the capacity.
Every dimension has a wall at both ends, so each loses cm:
The capacity — what it can hold — is the internal volume, cm.
Worked example 2 — the same box, open at the top. Now there is no lid, so the height loses only one thickness:
The open box uses less material and holds more, which is exactly what removing a lid should do. Ask where the material is before subtracting: two thicknesses for a closed pair of faces, one for an open end.
Worked example 3 — thicker walls. A closed wooden box has outer dimensions cm by cm by cm, and the wood is cm thick. Find the volume of wood.
Each dimension loses cm:
A sanity check worth doing: the wood is about a third of the box's outer volume, which is plausible for walls this thick relative to the size. If your answer came out larger than the external volume, you have added instead of subtracted.
Worked example 4 — from material to mass. If that wood has a mass of g per cubic centimetre, find the mass of the empty box.
Two surfaces exist for a hollow solid, and questions distinguish them. The external surface area is what you paint or polish; the internal surface area is what you line or waterproof. For the box in example 1, the internal dimensions give an internal total surface area of
against an external cm. Read whether the question wants the inside or the outside — it is worth several marks and no extra work.
Worked example 1 — a closed hollow box. A closed box has external dimensions cm by cm by cm and its walls are cm thick. Find the volume of material used and the capacity.
Every dimension has a wall at both ends, so each loses cm:
The capacity — what it can hold — is the internal volume, cm.
Worked example 2 — the same box, open at the top. Now there is no lid, so the height loses only one thickness:
The open box uses less material and holds more, which is exactly what removing a lid should do. Ask where the material is before subtracting: two thicknesses for a closed pair of faces, one for an open end.
Worked example 3 — thicker walls. A closed wooden box has outer dimensions cm by cm by cm, and the wood is cm thick. Find the volume of wood.
Each dimension loses cm:
A sanity check worth doing: the wood is about a third of the box's outer volume, which is plausible for walls this thick relative to the size. If your answer came out larger than the external volume, you have added instead of subtracted.
Worked example 4 — from material to mass. If that wood has a mass of g per cubic centimetre, find the mass of the empty box.
Two surfaces exist for a hollow solid, and questions distinguish them. The external surface area is what you paint or polish; the internal surface area is what you line or waterproof. For the box in example 1, the internal dimensions give an internal total surface area of
against an external cm. Read whether the question wants the inside or the outside — it is worth several marks and no extra work.
How does cross-section times length solve beam, pipe and canal problems?
Find the area of the cross-section, convert everything to one unit, and multiply by the length or by the distance the material travels.
Worked example 1 — a beam. A wooden beam has a cross-section cm by cm and a length of m. Find its volume in cubic metres.
Convert first: cm m and cm m.
Check in centimetres: cm, and since m cm, that is m, as required. Two routes, one answer — and the conversion factor for volume is a million, not a hundred, because three lengths are involved.
Worked example 2 — a canal. Water runs through a canal m wide and m deep at a speed of km per hour. How much water passes a point in one hour?
In one hour the water travels km m, so the volume is a cross-section dragged along that distance:
Worked example 3 — irrigation from that canal. If this water is spread over a field to a depth of cm, what area can be irrigated in one hour?
The volume is the same water, now a thin slab:
The volume is conserved and only its shape changes — a long thin channel of water becomes a wide thin sheet. That single idea answers almost every question in this section.
Worked example 4 — filling a tank through a pipe. A rectangular tank m by m is filled by a pipe of cross-section cm by cm through which water flows at m per second. How long does it take to fill the tank to a depth of m?
The volume needed is
The pipe delivers, each second,
Notice that the pipe's cross-section had to be in metres before being multiplied by a speed in metres per second. Mixing centimetres with metres here changes the answer by a factor of ten thousand, and it is the single commonest error in flow problems.
The general pattern. Whenever material moves at a steady speed through a fixed opening, the volume delivered in a time is
and every question of this kind — canal, pipe, conveyor, river — is that one line with different words.
Worked example 1 — a beam. A wooden beam has a cross-section cm by cm and a length of m. Find its volume in cubic metres.
Convert first: cm m and cm m.
Check in centimetres: cm, and since m cm, that is m, as required. Two routes, one answer — and the conversion factor for volume is a million, not a hundred, because three lengths are involved.
Worked example 2 — a canal. Water runs through a canal m wide and m deep at a speed of km per hour. How much water passes a point in one hour?
In one hour the water travels km m, so the volume is a cross-section dragged along that distance:
Worked example 3 — irrigation from that canal. If this water is spread over a field to a depth of cm, what area can be irrigated in one hour?
The volume is the same water, now a thin slab:
The volume is conserved and only its shape changes — a long thin channel of water becomes a wide thin sheet. That single idea answers almost every question in this section.
Worked example 4 — filling a tank through a pipe. A rectangular tank m by m is filled by a pipe of cross-section cm by cm through which water flows at m per second. How long does it take to fill the tank to a depth of m?
The volume needed is
The pipe delivers, each second,
Notice that the pipe's cross-section had to be in metres before being multiplied by a speed in metres per second. Mixing centimetres with metres here changes the answer by a factor of ten thousand, and it is the single commonest error in flow problems.
The general pattern. Whenever material moves at a steady speed through a fixed opening, the volume delivered in a time is
and every question of this kind — canal, pipe, conveyor, river — is that one line with different words.
Exam tip
What layout keeps a solids question free of unit errors?
Convert every length to one unit on the first line, and write the unit on every answer. In this chapter the arithmetic is straightforward and the units are where marks are lost.
- Pick a unit and convert before any multiplication. Volume conversions are cubes: m cm, and litre cm
- List which faces the question wants before choosing a formula: closed box six, open box five, walls and ceiling five, walls alone four
- **Use TSA (missing face) for an open box rather than adding five areas
- Subtract twice the thickness from each dimension of a hollow solid, and only once where an end is open
- Say whether a surface area is internal or external, and use the matching dimensions
- Apply the rate as the last step, and write the currency symbol in the answer
- Check TSA LSA on any cuboid; it costs one line and catches a mis-sorted pair of faces
- Subtract doors and windows from the wall area, not from the total
The misconception to name. Doubling every edge of a cube does not double its volume — it multiplies it by eight**, while the surface area goes up four times. A cube of edge cm has volume cm and surface cm; one of edge cm has volume cm and surface cm. Lengths scale once, areas twice, volumes three times — and a question comparing two similar solids is almost always testing that.
- Pick a unit and convert before any multiplication. Volume conversions are cubes: m cm, and litre cm
- List which faces the question wants before choosing a formula: closed box six, open box five, walls and ceiling five, walls alone four
- **Use TSA (missing face) for an open box rather than adding five areas
- Subtract twice the thickness from each dimension of a hollow solid, and only once where an end is open
- Say whether a surface area is internal or external, and use the matching dimensions
- Apply the rate as the last step, and write the currency symbol in the answer
- Check TSA LSA on any cuboid; it costs one line and catches a mis-sorted pair of faces
- Subtract doors and windows from the wall area, not from the total
The misconception to name. Doubling every edge of a cube does not double its volume — it multiplies it by eight**, while the surface area goes up four times. A cube of edge cm has volume cm and surface cm; one of edge cm has volume cm and surface cm. Lengths scale once, areas twice, volumes three times — and a question comparing two similar solids is almost always testing that.
Did you know
Why do small ice cubes melt faster than one big block?
Melting happens at the surface, while the amount to be melted is the volume. So what matters is the ratio of surface area to volume — and that ratio depends on size in a way that catches most people out.
For a cube of edge the ratio is
So it falls as the cube gets bigger:
- edge cm: ratio per cm
- edge cm: ratio per cm
- edge cm: ratio per cm
A small cube carries ten times as much surface per unit of volume as a cube ten times its edge. Break a block into small cubes and the total volume is unchanged while the total surface multiplies — so the ice meets far more warm air and melts much faster. It is the same reason sugar dissolves faster when powdered, why kindling catches before a log, and why cooking oil heats a thin strip of vegetable faster than a thick one.
The same arithmetic sets a limit on living things. A cell takes in food and oxygen through its surface and uses them throughout its volume, so as it grows the supply route falls behind the demand. That is why cells stay microscopic and divide instead of growing, and why a large animal needs lungs and intestines folded into enormous surfaces rather than a smooth skin.
And there is a shape question hiding here too. Among all cuboids of a fixed volume, which has the least surface? Take a volume of cm:
- : surface cm
- : surface cm
- : surface cm
The cube wins, and by a wide margin over the long thin box. It is the three-dimensional version of the result from the previous chapter, where the square beat every other rectangle of the same perimeter — and it is why a water tank, a packing carton and an igloo all tend towards the most compact shape available.
For a cube of edge the ratio is
So it falls as the cube gets bigger:
- edge cm: ratio per cm
- edge cm: ratio per cm
- edge cm: ratio per cm
A small cube carries ten times as much surface per unit of volume as a cube ten times its edge. Break a block into small cubes and the total volume is unchanged while the total surface multiplies — so the ice meets far more warm air and melts much faster. It is the same reason sugar dissolves faster when powdered, why kindling catches before a log, and why cooking oil heats a thin strip of vegetable faster than a thick one.
The same arithmetic sets a limit on living things. A cell takes in food and oxygen through its surface and uses them throughout its volume, so as it grows the supply route falls behind the demand. That is why cells stay microscopic and divide instead of growing, and why a large animal needs lungs and intestines folded into enormous surfaces rather than a smooth skin.
And there is a shape question hiding here too. Among all cuboids of a fixed volume, which has the least surface? Take a volume of cm:
- : surface cm
- : surface cm
- : surface cm
The cube wins, and by a wide margin over the long thin box. It is the three-dimensional version of the result from the previous chapter, where the square beat every other rectangle of the same perimeter — and it is why a water tank, a packing carton and an igloo all tend towards the most compact shape available.
Exam relevance
How are volume and surface area used in JEE and NEET questions?
This is foundation work that shows up more often in Physics and Chemistry numericals than in a Mathematics paper.
Where it leads in Mathematics. Class 10 adds the cylinder, cone and sphere and then combinations of them, all built on the same two ideas: cross-section times length for volume, and counting surfaces for area. The scaling result — lengths once, areas twice, volumes three times — becomes the ratio theorem for similar solids.
Where it leads in Physics. Cross-sectional area is the quantity in almost every formula that involves matter: density , pressure , resistance , and the equation of continuity for a flowing liquid — which is precisely your canal and pipe calculation restated. Both JEE Main and NEET set numericals where the physics is one line and the volume arithmetic is the rest.
Where the surface-to-volume ratio leads. It is examined directly in NEET Biology — cell size, alveoli, root hairs, villi — and in Chemistry when discussing the rate of a reaction with a powdered solid. **The formula you derived above is the quantitative form of an argument those papers make in words.
Question types to expect. At this level: direct volumes, surface areas, costs, hollow solids and flow problems. In competitive papers: continuity and flow-rate numericals, resistance of a wire of given dimensions, density and buoyancy, and assertion-reason items on scaling.
The single trap that costs marks.** Unit conversion. A cross-section in cm multiplied by a speed in m/s gives an answer wrong by a factor of ten thousand, and in a competitive paper that number will be among the options. Convert to SI units on the first line, always — the habit is worth more than any formula in this chapter.
A second trap. Confusing capacity with material. The inside volume is what a vessel holds; the difference between outside and inside is what it is made of. Chemistry questions about a container's contents want the first; density questions about the container itself want the second.
Board versus competitive emphasis. ICSE marks the formula, the face count, the unit and the cost; a competitive paper marks one number inside a longer problem. **The transferable sentence is volume is cross-section times length** — it carries you through cylinders, flow rates and continuity without a new idea.
Where it leads in Mathematics. Class 10 adds the cylinder, cone and sphere and then combinations of them, all built on the same two ideas: cross-section times length for volume, and counting surfaces for area. The scaling result — lengths once, areas twice, volumes three times — becomes the ratio theorem for similar solids.
Where it leads in Physics. Cross-sectional area is the quantity in almost every formula that involves matter: density , pressure , resistance , and the equation of continuity for a flowing liquid — which is precisely your canal and pipe calculation restated. Both JEE Main and NEET set numericals where the physics is one line and the volume arithmetic is the rest.
Where the surface-to-volume ratio leads. It is examined directly in NEET Biology — cell size, alveoli, root hairs, villi — and in Chemistry when discussing the rate of a reaction with a powdered solid. **The formula you derived above is the quantitative form of an argument those papers make in words.
Question types to expect. At this level: direct volumes, surface areas, costs, hollow solids and flow problems. In competitive papers: continuity and flow-rate numericals, resistance of a wire of given dimensions, density and buoyancy, and assertion-reason items on scaling.
The single trap that costs marks.** Unit conversion. A cross-section in cm multiplied by a speed in m/s gives an answer wrong by a factor of ten thousand, and in a competitive paper that number will be among the options. Convert to SI units on the first line, always — the habit is worth more than any formula in this chapter.
A second trap. Confusing capacity with material. The inside volume is what a vessel holds; the difference between outside and inside is what it is made of. Chemistry questions about a container's contents want the first; density questions about the container itself want the second.
Board versus competitive emphasis. ICSE marks the formula, the face count, the unit and the cost; a competitive paper marks one number inside a longer problem. **The transferable sentence is volume is cross-section times length** — it carries you through cylinders, flow rates and continuity without a new idea.
Key takeaways
What should you be able to calculate about solids before moving on?
One idea generates every formula here, and the units do the rest of the work.
- **Volume area of cross-section length**, which gives for a cuboid and for a cube
- **TSA and LSA , with LSA meaning the four walls only
- Cube**: , TSA , LSA , diagonal
- **Cuboid diagonal , which is the Pythagoras theorem used twice
- Check TSA LSA on every cuboid
- Count the faces the question names: six closed, five open or for walls-and-ceiling, four for walls alone — and subtract doors and windows from the walls
- Hollow solids: subtract twice the thickness from each dimension, once where an end is open; capacity is the inside, material is the difference
- Flow problems**: volume cross-section speed time, with everything in one unit
- Convert before multiplying: m cm and litre cm
- Scaling: double the edge and the area goes up four times, the volume eight
The best self-test is the hollow box. Take outer dimensions cm by cm by cm with walls cm thick, find the volume of material, and then work out how the answer changes if the box has no lid.
- **Volume area of cross-section length**, which gives for a cuboid and for a cube
- **TSA and LSA , with LSA meaning the four walls only
- Cube**: , TSA , LSA , diagonal
- **Cuboid diagonal , which is the Pythagoras theorem used twice
- Check TSA LSA on every cuboid
- Count the faces the question names: six closed, five open or for walls-and-ceiling, four for walls alone — and subtract doors and windows from the walls
- Hollow solids: subtract twice the thickness from each dimension, once where an end is open; capacity is the inside, material is the difference
- Flow problems**: volume cross-section speed time, with everything in one unit
- Convert before multiplying: m cm and litre cm
- Scaling: double the edge and the area goes up four times, the volume eight
The best self-test is the hollow box. Take outer dimensions cm by cm by cm with walls cm thick, find the volume of material, and then work out how the answer changes if the box has no lid.