X Tables
Maths History

Archimedes and the bathtub problem

Use a funny historical story to introduce volume, floating, measuring and scientific curiosity.

Quick answer

Archimedes discovered how to measure the volume of irregular objects by using water displacement – when an object is placed in water, it pushes water out of the way, and the amount of water that moves is equal to the volume of the object.

Learning outcome

By the end of this page, you will understand how to measure the volume of any object, even if it is not a simple shape like a cube or rectangle, by using water displacement. You will be able to:
  • Use a container of water to find the volume of an object
  • Explain why water moves when an object is placed in it
  • Apply this method to real-life situations like measuring toys or food
  • Understand how scientists use experiments to discover new things

Why this matters

This story shows how curiosity and a good idea can help us solve problems we didn’t think we could solve. The method Archimedes used is still used today by scientists, engineers, and even in your kitchen when you measure ingredients. You can use this method to find out how much space something takes up, no matter how strange its shape is. It’s like being a detective with water!

How it works

Let’s imagine you have a strange-shaped toy that isn’t a cube or a rectangle. You want to know how much space it takes up. You can do this by:
  1. Fill a container with water
  2. Mark the water level
  3. Put the object in the water
  4. Notice how the water level rises
  5. Measure the difference in water level
The amount of water that moved is the volume of your object.

Examples you can see

Let’s try this with some simple examples: Example 1: A toy car Imagine you have a toy car and want to know how much space it takes up. You take a large glass, fill it with water, and mark the water level. Then you put the car in the water. The water level goes up. You measure how much higher the water is now compared to before. That difference is the volume of your car. Example 2: A potato If you are cooking and want to know how big a potato is, you can do the same thing. Fill a bowl with water, drop the potato in, and see how much water is pushed out. That’s how much space the potato takes up. Example 3: A small stone You can even use this to find the volume of a small stone. If you drop it in a measuring jug, the water will rise. You can read the new level and subtract the old level to find the volume of the stone.

Try it as a mini adventure

Adventure: The Great Water Challenge You are a scientist, and your mission is to find the volume of three different objects: a toy duck, a marble, and a small book. You have a measuring jug, a container of water, and a notebook to write down your results.
  1. Fill the measuring jug with 100 ml of water
  2. Put the toy duck in the water and watch what happens
  3. Write down the new water level
  4. Remove the duck and write down the original level
  5. Calculate the difference
  6. Repeat for the marble and the book
What do you notice? Does the duck take up more or less space than the marble? What about the book? The more space an object takes up, the more water it pushes out.

Common mistakes

  • Mistake: Not marking the water level before putting the object in.
    Why it happens: You forget to take a note of the starting water level.
    Fix: Always mark or remember the water level before adding the object.
  • Mistake: Thinking the water level rises because the object is heavy.
    Why it happens: Children sometimes confuse weight with volume.
    Fix: Explain that volume is about space, not weight. A small stone pushes out more water than a big feather because it takes up more space.
  • Mistake: Not fully submerging the object.
    Why it happens: The object floats or is too big.
    Fix: Use a tool to push it down or choose a smaller object that fits in the container.
  • Mistake: Reading the water level wrong.
    Why it happens: Looking at the water from an angle or not reading the scale carefully.
    Fix: Look at the water level from the front and read the scale carefully.

Parent and carer guide

To help your child understand this concept, you can:
  • Use a clear container like a measuring jug or glass
  • Fill it with water and let your child mark the water level
  • Have them place different objects in the water and observe what happens
  • Ask questions like: “What do you notice?” or “Which object pushed out more water?”
  • Encourage them to write down or draw their results
  • Keep the activity fun and exploratory, not like a test
What to say: - “Let’s see how much space this toy takes up.” - “How much water did it push out?” - “Which object do you think will take up the most space?” What not to worry about: - Don’t worry if they get the exact measurement wrong. The idea is to understand the process. - It’s okay if they don’t get the same answer twice – this is part of science! - Keep the focus on curiosity, not perfect results.

Practice questions

  1. Why does water rise when you put an object in it?
  2. If a stone pushes out 50 ml of water, what is its volume?
  3. How can you use water to find the volume of a toy car?
  4. Which object would push out more water: a small ball or a big ball?
  5. Why is it important to mark the water level before adding the object?
Answers:
  1. Because the object takes up space and pushes the water out of the way.
  2. Its volume is 50 ml.
  3. You put the car in water, mark the water level, add the car, and see how much it rises.
  4. The big ball would push out more water because it takes up more space.
  5. So you know exactly how much water was moved by the object.

FAQ

What is the story of Archimedes and the bathtub?

The story says that Archimedes was asked to find out if a crown was made of pure gold. He couldn’t weigh it or cut it open. While taking a bath, he noticed the water rose when he got in. This made him realize he could use water displacement to measure volume and solve the problem.

Why is volume important?

Volume tells us how much space an object takes up. It helps us know how much liquid fits in a container, how much space a toy needs, or how much material we need to build something.

Can you measure volume without water?

Yes! You can use other methods like filling a container with sand or rice, or even by calculating the size of a regular shape like a box or a ball.

How do scientists use this today?

Scientists use water displacement to measure the volume of rocks, animals, and even small pieces of art. It’s a simple but very useful way to find out how much space something takes up.

Is this method accurate for all objects?

It works for most solid objects that don’t float or dissolve in water. For objects that float, you can use a tool to push them down or use a different method.