Volume and Measuring by Displacement വ്യാപ്തം
When an object goes into water, it pushes some water out of the way. How much? Exactly its own volume – the space it takes up. This idea is the heart of Archimedes' principle. On this page you will learn the units of volume (cm³, m³, litre, mL) and how to work out the volume of a box and a ball. You will also learn the clever trick for finding the volume of an odd-shaped stone: put it in water and see how much water it displaces.
- Multiplying three numbers; powers of ten such as 10³ and 10⁶.
- Mass and weight (page 1) – only for the last section.
1. What is volume?
The volume (വ്യാപ്തം) of an object is the amount of space it occupies. A big jackfruit takes up more space than a mango, so it has a bigger volume. We measure volume by counting how many small unit cubes would fill the same space. A cube 1 cm long, 1 cm wide and 1 cm high has a volume of 1 cubic centimetre (1 cm³).
For a box shape (a cuboid), the number of unit cubes is simply length × breadth × height:
Count the cubes in one layer (l × b), then multiply by the number of layers (h). That is why V = l × b × h. The answer is in cm³ because cm × cm × cm = cm³.
2. Units of volume
| Unit | What it is | Link to the others |
|---|---|---|
| cubic centimetre, cm³ | a cube of side 1 cm (about the size of a sugar cube) | 1 cm³ = 1 mL |
| millilitre, mL | the unit on medicine cups and measuring cylinders | 1000 mL = 1 L |
| litre, L | a cube of side 10 cm; a bottle of drinking water | 1 L = 1000 cm³ |
| cubic metre, m³ (SI unit) | a cube of side 1 m (100 cm) | 1 m³ = 100 × 100 × 100 cm³ = 10⁶ cm³ = 1000 L |
1 mL = 1 cm³ · 1 L = 1000 cm³ · 1 m³ = 10⁶ cm³ = 1000 L. To change cm³ into m³, divide by 10⁶ (move the point 6 places left): 100 cm³ = 100 ÷ 1 000 000 m³ = 1 × 10⁻⁴ m³.
A Milma milk packet holds 500 mL = 500 cm³. A ration-shop measure of coconut oil holds 1 L = 1000 cm³. The black 1000 L water tank on the roof of many Kerala houses holds 1 m³. A teaspoon holds about 5 mL.
Worked example 1 · A wooden block
Q. A wooden block is 8 cm long, 5 cm wide and 2.5 cm high. Find its volume in cm³, in mL and in m³.
V = l × b × h gives cm³. 1 cm³ = 1 mL. Divide by 10⁶ for m³.
3. The volume of a ball (sphere)
A ball shape is called a sphere. Its size is given by its radius r (centre to surface) or its diameter d = 2r. You will prove its volume formula in higher classes; for now, just use it:
Notice that the formula has only r in it. It does not ask what the sphere is made of. So two spheres with the same diameter have the same volume, whether they are copper, iron, wood or wax. This is the key to question 2 of “Let us assess” (page 11).
Make both diameters 6 cm: the volumes become exactly equal (113.04 cm³ with π = 3.14), even though copper is a different metal from iron. Doubling the diameter makes the volume 2 × 2 × 2 = 8 times bigger.
Worked example 2 · Volume of a ball
Q. Find the volume of an iron ball of radius 3 cm. (Take π = 3.14.)
r³ = 3 × 3 × 3 = 27. Then multiply by 4, by 3.14 and divide by 3.
4. Odd shapes: measure by displacement
A stone or a key has no simple formula. But water has no fixed shape: it flows around anything. When you lower an object into water, the object takes up some space, and the water that was there has to go somewhere – it is displaced (pushed aside). The volume of water displaced (ആദേശം ചെയ്ത ജലം) equals the volume of the part of the object under water. When the object is fully under water, it displaces water equal to its whole volume.
Method 1 · Measuring cylinder
Pour some water into a measuring cylinder and read the level. Tie the object with a thread and lower it in. The level rises. Volume of the object = new reading − old reading. Read at the bottom of the curved water surface, with your eye level with it.
Half the stone under water raises the level by half its volume. Fully under, the rise is the whole volume. The water rises by the same amount whatever the object is made of; only the size of the part under water matters.
Method 2 · Overflow jar
Fill an overflow jar (കവിഞ്ഞൊഴുകും ജാർ), a can with a spout near the top, until water just starts to drip from the spout. Wait until it stops. Put an empty beaker under the spout and lower the object in. The overflowing water (കവിഞ്ഞൊഴുകിയ ജലം) collected in the beaker has exactly the same volume as the object. This is the method used in the textbook's Archimedes activity (Fig. 5.6, page 7 of this guide).
The water level in the jar does not rise – it is already at the spout – so every bit of water pushed aside pours into the beaker.
Worked example 3 · Volume of a stone
Q. A measuring cylinder contains water up to the 60 mL mark. When a stone is fully lowered into it, the water rises to 85 mL. Find the volume of the stone in cm³ and in m³.
Volume of the stone = rise in the water level. 1 mL = 1 cm³, and 1 cm³ = 10⁻⁶ m³.
5. Archimedes and the golden crown
About 2300 years ago, King Hiero of Syracuse asked Archimedes to check whether his new golden crown had cheaper metal mixed in. To find out, Archimedes needed the crown's volume – but he was not allowed to melt or damage it. One day, stepping into a full bathtub, he saw the water overflow and realised: an object pushes out water equal to its own volume. That is exactly the overflow-jar method above. You will finish this story on page 7.
An object (or the part of it) under water displaces water equal to its own volume. Volume of an odd shape = rise in level of a measuring cylinder = volume of water that overflows from a full overflow jar. Equal volumes under water → equal volumes of water displaced.