E. Spatial Sense · Spatial Sense: 3D Objects, Transformations, Circles, Surface Area and VolumeVolume and Capacity
E2.1About 45 minutes
Learning goals- I can describe how volume and capacity are alike and how they are different.
- I can use the relationship 1 cm³ = 1 mL to change between volume and capacity units.
- I can find the volume of a rectangular prism by thinking about layers of cubes.
- I can measure the volume of an object by displacement.
Words to know
volume
The amount of space an object takes up. Measured in cubic units such as cubic centimetres (cm³) and cubic metres (m³).
capacity
The amount a container can hold. Usually measured in millilitres (mL) and litres (L).
cubic centimetre (cm³)
The volume of a cube that is 1 cm long, 1 cm wide and 1 cm high.
cubic metre (m³)
The volume of a cube that is 1 m on each side.
displacement
Pushing water out of the way. An object put under water raises the water level by its own volume.
Remember from earlier grades
In earlier grades you measured capacity, and in Grade 4 you learned how millilitres and litres are related. In Grade 5 and Grade 6 you converted metric units, such as 2.5 L = 2500 mL. In Grade 6 you found the surface area of prisms: the area that covers the outside. Now you'll measure the space inside.
Let's start with a juice box
A juice box says 200 mL on the side. That's its capacity: how much juice it holds. You could also measure how much space the box takes up in your lunch bag. That's its volume. Are these the same thing? Almost! Today you'll find out exactly how they're connected.
Alike and different
Volume is the amount of space an object takes up. Any object has a volume: a rock, a book, a juice box. We measure volume in cubic units, like cm³ (cubic centimetres) and m³ (cubic metres).
Capacity is the amount a container can hold. We usually measure capacity in mL and L.
Alike: both measure 3D space. A container's capacity is the volume of the space inside it.
Different:
- A solid rock has a volume but no capacity: it can't hold anything.
- A container with thick walls has a capacity that is less than its outside volume, because the walls take up space.
The key connection
A hollow cube that is 1 cm on each side holds exactly 1 mL of water.
1 cm³ = 1 mL
So a cube that is 10 cm on each side has a volume of 10 × 10 × 10 = 1000 cm³ and holds 1000 mL = 1 L.
A cube that is 1 m on each side has a volume of 100 cm × 100 cm × 100 cm = 1 000 000 cm³. That's 1 000 000 mL = 1000 L. So 1 m³ = 1000 L.
Volume of a rectangular prism
Imagine filling a box with 1 cm cubes. The bottom layer of a box 5 cm long and 4 cm wide holds 5 × 4 = 20 cubes. If the box is 3 cm tall, there are 3 layers: 20 × 3 = 60 cubes.
Volume = length × width × height = 5 × 4 × 3 = 60 cm³. You'll see at the end of this unit that this is really area of the base × height, and that it works for every prism and cylinder.
Worked exampleExample 1: From volume to capacity
The inside of a juice box measures 4 cm by 6 cm by 10 cm. What is its capacity?
- Volume of the inside: 4 × 6 × 10 = 240 cm³.
- 1 cm³ = 1 mL, so 240 cm³ = 240 mL.
Worked exampleExample 2: Displacement
A measuring cup has 300 mL of water. After a stone is dropped in and is completely under water, the water reads 375 mL. What is the volume of the stone?
- The stone pushed the water up by 375 − 300 = 75 mL.
- The water went up by exactly the stone's volume. 75 mL = 75 cm³.
The stone's volume is 75 cm³.
Worked exampleExample 3: An aquarium
An aquarium is 50 cm long, 30 cm wide and 40 cm tall. How many litres does it hold when it is full?
- Volume = 50 × 30 × 40 = 60 000 cm³.
- 60 000 cm³ = 60 000 mL.
- 1000 mL = 1 L, so 60 000 mL ÷ 1000 = 60 L.
Worked exampleExample 4: Thick walls
A wooden box measures 20 cm by 20 cm by 20 cm on the outside. Its walls, bottom and lid are each 1 cm thick. Compare its volume and its capacity.
- Outside volume: 20 × 20 × 20 = 8000 cm³.
- Inside, each measurement loses 1 cm on each side: 20 − 2 = 18 cm.
- Inside space: 18 × 18 × 18 = 5832 cm³, so the capacity is 5832 mL, or about 5.8 L.
Volume 8000 cm³; capacity 5832 mL. The capacity is less, because the wood takes up space.
Which word?
Ask yourself: Am I asking how much space it takes up, or how much it can hold?
- "How much soup does this pot hold?" → capacity (mL, L)
- "How much space does this brick take up?" → volume (cm³, m³)
And remember the bridge between them: 1 cm³ = 1 mL.
Just for funWhy can't a rock be a juice box?
It has plenty of volume, but zero capacity.
Hands-on activityMeasure by displacement
You need: a clear measuring cup with mL markings (a plastic one is best), water, and small objects that sink and that can get wet: a stone, a metal spoon, a handful of marbles, a small toy.
- Fill the cup to an easy line, like 300 mL. Read it at eye level.
- Gently put one object in so it is completely under water. Read the new level.
- Subtract to find the object's volume in mL, then write it in cm³.
- Find a small plastic container with straight sides and a flat bottom (like a food-storage box). Measure the inside with a ruler and calculate its capacity. Then fill it with water from the measuring cup to check. Was your calculation close?
What to look for: the rise in mL equals the volume in cm³. Do this over a sink or a towel.
Your turn
3000 cm³. 3 L = 3000 mL = 3000 cm³.
3What is the capacity of a box whose inside measures 5 cm by 4 cm by 3 cm?
60 mL. 5 × 4 × 3 = 60 cm³ = 60 mL.
4Water rises from 400 mL to 465 mL when a toy is dropped in. What is the toy's volume?
65 cm³. 465 − 400 = 65 mL = 65 cm³.
5Does a solid rock have a capacity? Explain.
No. It has a volume (it takes up space), but it has no empty space inside to hold anything.
6How many litres are in 1 m³?
1000 L. 1 m³ = 100 × 100 × 100 = 1 000 000 cm³ = 1 000 000 mL, and 1 000 000 ÷ 1000 = 1000 L.
7Volume or capacity: the amount of soup a pot can hold?
More practice: warm-up
0.5 L. 500 cm³ = 500 mL = 0.5 L.
5Which unit would you use for the volume of a brick: cm³ or L?
cm³ (a brick is solid, so we measure the space it takes up).
More practice: on your own
1A container's inside measures 10 cm by 10 cm by 25 cm. How many litres does it hold?
2.5 L. 10 × 10 × 25 = 2500 cm³ = 2500 mL = 2.5 L.
2A thermos holds 0.35 L. What is its capacity in cm³?
350 cm³. 0.35 L = 350 mL = 350 cm³.
38 identical marbles raise the water in a cup from 200 mL to 240 mL. What is the volume of one marble?
5 cm³. Together: 240 − 200 = 40 cm³. One marble: 40 ÷ 8 = 5 cm³.
4An aquarium is 60 cm long, 30 cm wide and 35 cm tall. It is filled to 5 cm below the top. How many litres of water are in it?
54 L. The water is 35 − 5 = 30 cm deep. 60 × 30 × 30 = 54 000 cm³ = 54 000 mL = 54 L.
5A box is 12 cm by 10 cm by 8 cm on the outside, with walls 1 cm thick everywhere. What is its capacity?
480 mL. Each inside measurement is 2 cm less: 10 × 8 × 6 = 480 cm³ = 480 mL. (Its outside volume is 960 cm³, double the capacity!)
Challenge
1A rain barrel holds 0.2 m³ of water. How many litres is that? How many times could you fill a 4 L watering can from it?
200 L and 50 times. 1 m³ = 1000 L, so 0.2 m³ = 200 L. 200 ÷ 4 = 50.
2A baking pan is 30 cm by 20 cm. You pour in 1.5 L of water. How deep is the water?
2.5 cm. 1.5 L = 1500 cm³. The bottom has an area of 30 × 20 = 600 cm². Depth = 1500 ÷ 600 = 2.5 cm.
3A recipe needs 0.75 L of broth. You have a container whose inside measures 8 cm by 8 cm by 12 cm. Is it big enough? Explain.
Yes. 8 × 8 × 12 = 768 cm³ = 768 mL. 0.75 L = 750 mL. 768 mL is more than 750 mL, with 18 mL to spare.
Quiz
4. The water in a measuring cup rises from 120 mL to 170 mL when a rock is added. What is the rock's volume?
5. What is the volume of a box 2 cm by 5 cm by 10 cm?
Big idea: E2.1 asks students to describe the differences and similarities between volume and capacity, and to use the relationship between millilitres and cubic centimetres to solve problems.
How to help at home: Read labels together ("This can holds 540 mL. How many cm³ is that?"). Do the displacement activity with a measuring cup. Ask your child to explain why a thick mug holds less than its outside size suggests.
Common difficulties:
- Thinking volume and capacity are unrelated. The bridge is 1 cm³ = 1 mL.
- Converting 1 m³ to 1 000 cm³ or 100 L. A cubic metre is 100 × 100 × 100 = 1 000 000 cm³ = 1000 L. The next lesson practises these conversions.
- Reading the measuring cup from above. Read at eye level for an accurate measurement.
Safety: Use a plastic measuring cup and do water activities over a sink or towel. Keep marbles and other small objects away from younger children, who could choke on them.