Hello, Super Mathematicians! Welcome to the World of Volume!
Have you ever wondered how much space a toy car, a funny-shaped rock, or your favourite action figure takes up? It's easy to find the space (or volume) of a square box, but what about lumpy, bumpy things? That's what we're going to learn today!
In these notes, you will become a detective and learn a super cool trick called the water displacement method to measure the volume of any small object, no matter its shape. It’s like magic, but it’s actually science and maths working together!
1. Quick Review: What are Volume and Capacity?
Before we learn the new trick, let's remember a few things.
What is Volume?
Volume is the amount of 3D space an object takes up. Think about a small sugar cube. The space it fills is its volume. We often measure the volume of solids in cubic centimetres ($$\text{cm}^3$$) or cubic metres ($$\text{m}^3$$).
For example, a cube with sides of 1 cm has a volume of $$1 \text{ cm} \times 1 \text{ cm} \times 1 \text{ cm} = 1 \text{ cm}^3$$.
What is Capacity?
Capacity is the amount a container can hold. Think about your water bottle. The amount of water it can hold when it's full is its capacity. We often measure capacity in millilitres (mL) and litres (L).
2. The Magical Link: Connecting Volume and Capacity
Here's the most important secret for today! Volume and capacity are related in a very special way, especially when we talk about water.
The Golden Rule:
An object with a volume of 1 cubic centimetre ($$1 \text{ cm}^3$$) will push away, or displace, exactly 1 millilitre ($$1 \text{ mL}$$) of water.
So, we can say:
$$1 \text{ cm}^3 = 1 \text{ mL}$$
And because there are 1000 mL in a litre:
$$1000 \text{ cm}^3 = 1000 \text{ mL} = 1 \text{ L}$$
Quick Review Box
Remember this forever!
Volume of $$1 \text{ cm}^3$$ = Capacity of $$1 \text{ mL}$$
They are the same amount of space!
3. The Puzzle: Irregular Solids
A cube or a cuboid is a regular solid. We can find its volume by measuring its length, width, and height and multiplying them together.
But what about an irregular solid? This is an object that doesn't have straight, easy-to-measure sides.
Examples: A stone, a marble, a small potato, a toy car.
We can't use a ruler to find the volume of a lumpy rock. So, how do we solve this puzzle? With water!
4. The Solution: The Water Displacement Method
This sounds fancy, but the idea is simple. Have you ever gotten into a bathtub that was very full? What happened? The water level went up, and some might have even spilled out! The amount of water that moved up is equal to the volume of the part of your body that went into the water.
We use this same idea to find the volume of irregular solids. This method is called the water displacement method. ("Displacement" just means "pushed away").
There are two main ways to do this:
- Using a Measuring Cup or Tank
- Using an Overflow Vessel
Did you know?
The story goes that a famous ancient Greek scientist named Archimedes discovered this idea in his bathtub! He was so excited that he shouted "Eureka!" (which means "I have found it!") and ran out to share his discovery.
Method 1: Using a Measuring Cup (or Tank)
This is great for smaller objects. You just need a measuring cup with markings (in mL) and some water.
Step-by-Step Guide:
Step 1: Pour water into the measuring cup.
Pour enough water to cover the object, but not so much that it will spill. Let's say you pour in 200 mL.
Step 2: Record the starting water level.
Look at the markings on the cup carefully. This is your 'before' measurement.
Example: Starting water level = 200 mL
Step 3: Gently place the irregular solid into the water.
Make sure the object is completely under the water. Don't drop it in and splash! The water level will rise.
Step 4: Record the new water level.
This is your 'after' measurement.
Example: The water level rises to 250 mL. New water level = 250 mL
Step 5: Find the difference!
Subtract the starting water level from the new water level. The difference is the volume of the water that was displaced.
$$ \text{New water level} - \text{Starting water level} = \text{Volume of water displaced} $$
Example: $$250 \text{ mL} - 200 \text{ mL} = 50 \text{ mL}$$
Step 6: Convert to the correct units.
Remember our Golden Rule? $$1 \text{ mL} = 1 \text{ cm}^3$$. So, the volume of the object is the same number, but in $$cm^3$$.
Example: The volume of the object is 50 $$cm^3$$.
Key Takeaway for Method 1
The volume of the object is the amount the water level went up.
Method 2: Using an Overflow Vessel
What if the object is a bit bigger? We can use a special container called an overflow vessel. It looks like a can with a little spout on the side.
Step-by-Step Guide:
Step 1: Prepare the overflow vessel.
Place the overflow vessel on a flat surface. Fill it with water until the water starts to drip out of the spout. Wait for it to stop dripping completely. Now the water is exactly at the level of the spout.
Step 2: Place an empty measuring cup under the spout.
This cup will catch all the water that spills, or "overflows". Make sure the cup is empty!
Step 3: Gently place the irregular solid into the vessel.
Lower the object in slowly. As it sinks, it will push water out through the spout and into your empty measuring cup.
Step 4: Wait for the water to stop dripping.
Be patient! Once the last drop has fallen into the measuring cup, you can move on.
Step 5: Measure the collected water.
Take the measuring cup that caught the overflow. The amount of water in it is the volume of the water displaced.
Example: You measure the water in the cup and find it is 85 mL.
Step 6: Convert to the correct units.
Again, use the Golden Rule: $$1 \text{ mL} = 1 \text{ cm}^3$$.
Example: The volume of the object is 85 $$cm^3$$.
Key Takeaway for Method 2
The volume of the object is equal to the volume of the water that spilled out.
5. Final Summary - You're a Volume Expert!
Let's remember the most important things we learned.
Volume is the space an object takes up ($$\text{cm}^3$$).
Capacity is how much a container can hold (mL, L).
The Magical Link: $$1 \text{ cm}^3 = 1 \text{ mL}$$. This is your superpower!
An irregular solid is an object with a funny shape that is hard to measure with a ruler.
The Water Displacement Method helps us find the volume of irregular solids.
Method 1 (Measuring Cup): Volume = New Water Level - Starting Water Level.
Method 2 (Overflow Vessel): Volume = Amount of Water Collected.
Great job! Finding volume this way is a fun experiment. You can even try it at home with a kitchen measuring jug and small waterproof toys (just make sure to ask for permission first!). Keep practicing, and you'll be a master of measuring everything!