How to Measure the Volume of a Stone With No Shape at All
Learn to convert area and volume units, calculate the area of squares, triangles and circles, find the area of a leaf on graph paper, and measure an irregular solid by water displacement.
How can you measure the volume of a stone that has no regular shape?
By dropping it into water and seeing how much the water level rises. The stone pushes aside exactly its own volume, so the rise is the answer — no formula required.
That idea, the displacement method, is the heart of this chapter. This page covers everything in the ICSE Class 7 Physics chapter's first half: area and its units, the area of regular and irregular shapes, volume and its units, and measuring the volume of liquids and irregular solids.
That idea, the displacement method, is the heart of this chapter. This page covers everything in the ICSE Class 7 Physics chapter's first half: area and its units, the area of regular and irregular shapes, volume and its units, and measuring the volume of liquids and irregular solids.
What is area and how do you convert between its units?
Area is the surface occupied by a body, and its SI unit is the square metre ().
Conversions follow from squaring the length conversion, which is why the numbers look large:
Worked example. A floor mat of area in square centimetres is
A school playground is the everyday case: a ground of is .
Here is the misconception that costs marks. Since , students write . It is , because both the length and the breadth are converted. Picture a square metre ruled into centimetre squares: there are 100 rows of 100 squares.
Conversions follow from squaring the length conversion, which is why the numbers look large:
Worked example. A floor mat of area in square centimetres is
A school playground is the everyday case: a ground of is .
Here is the misconception that costs marks. Since , students write . It is , because both the length and the breadth are converted. Picture a square metre ruled into centimetre squares: there are 100 rows of 100 squares.
Formula
What are the formulae for the area of regular shapes?
Each regular figure has its own standard formula:
Worked examples.
A rectangular blackboard by has area .
A triangular set-square with base and height has area
A circular plate of radius has area
For an irregular shape such as a leaf, use the graph-paper method: place the leaf on 1 mm graph paper, trace its outline, then count the squares inside. Count every complete square, count every square more than half enclosed, and ignore those half or less.
Suppose a traced leaf covers 24 complete centimetre squares and 10 squares more than half enclosed. Its area is about
The result is an estimate, not an exact value, and smaller squares give a better one. That is the boundary of the method: it trades exactness for the ability to measure a shape no formula fits.
Worked examples.
A rectangular blackboard by has area .
A triangular set-square with base and height has area
A circular plate of radius has area
For an irregular shape such as a leaf, use the graph-paper method: place the leaf on 1 mm graph paper, trace its outline, then count the squares inside. Count every complete square, count every square more than half enclosed, and ignore those half or less.
Suppose a traced leaf covers 24 complete centimetre squares and 10 squares more than half enclosed. Its area is about
The result is an estimate, not an exact value, and smaller squares give a better one. That is the boundary of the method: it trades exactness for the ability to measure a shape no formula fits.
What is volume and how do litres relate to cubic centimetres?
Volume is the space occupied by a body, and its SI unit is the cubic metre ().
The conversions you need are:
and since a millilitre is a thousandth of a litre, exactly. That equality is worth memorising, because it lets you move between a measuring cylinder's millilitres and a formula's cubic centimetres without any arithmetic.
Worked example. A water bottle holds , which is
A household water tank of holds 1000 litres — the same tank a plumber would call a 1000-litre tank.
For a regular solid, calculate rather than measure. A cubical box of side has volume .
As with area, cube the length factor: is , not .
The conversions you need are:
and since a millilitre is a thousandth of a litre, exactly. That equality is worth memorising, because it lets you move between a measuring cylinder's millilitres and a formula's cubic centimetres without any arithmetic.
Worked example. A water bottle holds , which is
A household water tank of holds 1000 litres — the same tank a plumber would call a 1000-litre tank.
For a regular solid, calculate rather than measure. A cubical box of side has volume .
As with area, cube the length factor: is , not .
How do you measure liquid volume and the volume of an irregular solid?
With a measuring cylinder, read at the lower meniscus with your eye level with the surface.
Water curves upward at the edges where it touches the glass, forming a meniscus. Take the reading at the bottom of that curve, and keep your eye exactly level with it — looking from above or below shifts the apparent reading, an error called parallax.
For an irregular solid such as a stone, use water displacement:
1. Pour water into the cylinder and note the level, say .
2. Lower the stone in gently, fully submerged, using a thread.
3. Note the new level, say .
4. The volume of the stone is the difference:
If the solid is too large for the cylinder, use an overflow can and collect the water that spills into a cylinder — the collected volume equals the solid's volume.
The method has two genuine limits. The solid must sink, so a cork has to be held down or tied to a heavier sinker. And it must not dissolve in or absorb the water, which is why a sugar cube or a dry sponge cannot be measured this way.
Water curves upward at the edges where it touches the glass, forming a meniscus. Take the reading at the bottom of that curve, and keep your eye exactly level with it — looking from above or below shifts the apparent reading, an error called parallax.
For an irregular solid such as a stone, use water displacement:
1. Pour water into the cylinder and note the level, say .
2. Lower the stone in gently, fully submerged, using a thread.
3. Note the new level, say .
4. The volume of the stone is the difference:
If the solid is too large for the cylinder, use an overflow can and collect the water that spills into a cylinder — the collected volume equals the solid's volume.
The method has two genuine limits. The solid must sink, so a cork has to be held down or tied to a heavier sinker. And it must not dissolve in or absorb the water, which is why a sugar cube or a dry sponge cannot be measured this way.
Exam tip
Exam tip: squaring and cubing the conversion factor
Most lost marks in this chapter are unit conversions, not physics.
Before converting, ask whether the quantity is a length, an area or a volume. Then use the factor once for length, squared for area and cubed for volume. So becomes for and for .
Write the unit beside every number at every step. An answer of "18" is incomplete; is the answer.
For displacement questions, always show the subtraction line — final level minus initial level — because that step carries the method mark even if the arithmetic slips.
And when counting squares for an irregular area, state your rule in the answer: complete squares counted, squares more than half enclosed counted, the rest ignored. Examiners award that reasoning separately from the number.
Before converting, ask whether the quantity is a length, an area or a volume. Then use the factor once for length, squared for area and cubed for volume. So becomes for and for .
Write the unit beside every number at every step. An answer of "18" is incomplete; is the answer.
For displacement questions, always show the subtraction line — final level minus initial level — because that step carries the method mark even if the arithmetic slips.
And when counting squares for an irregular area, state your rule in the answer: complete squares counted, squares more than half enclosed counted, the rest ignored. Examiners award that reasoning separately from the number.
Did you know
Why does a submerged stone push aside exactly its own volume?
Because two things cannot occupy the same space at the same time.
When the stone goes under, the water that was in that space has nowhere to go but upward. So the rise in level accounts for precisely the space the stone now fills — no more, no less.
That is why the method works for any shape at all, however knobbly. The water does the measuring, and it never needs to know what the shape is.
When the stone goes under, the water that was in that space has nowhere to go but upward. So the rise in level accounts for precisely the space the stone now fills — no more, no less.
That is why the method works for any shape at all, however knobbly. The water does the measuring, and it never needs to know what the shape is.
Key takeaways
Area, volume and measurement: quick revision
- Area is the surface occupied by a body, measured in ; volume is the space occupied, measured in .
- Square the factor for area () and cube it for volume ().
- and , so a measuring cylinder's millilitres are cubic centimetres already.
- Standard areas: square , rectangle , triangle , circle .
- For an irregular lamina, count complete squares plus those more than half enclosed on graph paper — an estimate, improved by smaller squares.
- Read a measuring cylinder at the lower meniscus at eye level, and find an irregular solid's volume as the rise in water level, provided it sinks and does not dissolve.
You will remember all of this far better after answering five questions on it than after reading it twice.
- Square the factor for area () and cube it for volume ().
- and , so a measuring cylinder's millilitres are cubic centimetres already.
- Standard areas: square , rectangle , triangle , circle .
- For an irregular lamina, count complete squares plus those more than half enclosed on graph paper — an estimate, improved by smaller squares.
- Read a measuring cylinder at the lower meniscus at eye level, and find an irregular solid's volume as the rise in water level, provided it sinks and does not dissolve.
You will remember all of this far better after answering five questions on it than after reading it twice.