A friendly flat-vector scene of a cube and a cuboid tank partly filled with water, in IllumiTutor navy and amber on an off-white background.

Volume of Cubes and Cuboids

How cubic units describe space inside solids, how base area and height work together, and how to reverse a volume calculation safely.

⏱ 6 min · 🎯 4 things to master

Area covers a flat shape. Volume measures the space inside a solid, such as a box, a cube or the water in a tank. We can fill the solid with equal layers, count the space in one layer, and then work out how many layers fit.

Parents: let your child read the givens and predict the missing dimension before opening a calculation. The lab shows the same solid changing between a question state and a checked state.

By the end you'll be able to find cube and cuboid volume, use base area times height, use the water height in a tank, and reverse a volume calculation with division or a recognised root.

Three lengths make cubic volume

is the space inside a solid. A has length, width and height. A has the same edge length in all three directions.

For a cuboid, multiply the three lengths. For a cube, multiply the edge three times. The unit is cubic because three length units are multiplied.

Build the volume

Cuboid volume=length×width×height\text{Cuboid volume}=\text{length}\times\text{width}\times\text{height}
5 cm×4 cm×3 cm=60 cm35\,\text{cm}\times4\,\text{cm}\times3\,\text{cm}=60\,\text{cm}^3
Cube volume=2 cm×2 cm×2 cm=8 cm3\text{Cube volume}=2\,\text{cm}\times2\,\text{cm}\times2\,\text{cm}=8\,\text{cm}^3

🤔 Predict first: What is the volume of a cuboid 6 cm long, 3 cm wide and 2 cm high?

See the same formula as base area times height

The bottom face of a cuboid is its . Find the base area first, then stack that rectangle to the given height. This gives the equivalent rule volume = base area × height.

Stack equal layers

base area=12 cm2\text{base area}=12\,\text{cm}^2
volume=12 cm2×5 cm=60 cm3\text{volume}=12\,\text{cm}^2\times5\,\text{cm}=60\,\text{cm}^3

🤔 Predict first: A cuboid has a base area of 20 cm² and a height of 4 cm. What is its volume?

Find liquid volume from the water height

Water in a cuboid tank takes the shape of a cuboid. Use the height the water reaches, not the full tank height. A useful conversion is 1 L = 1000 cm³.

🤔 Predict first: A tank base is 20 cm by 10 cm. Water reaches 6 cm high. What is the water volume?

Show the worked solution

Check the water level calculation

water volume=20 cm×10 cm×6 cm\text{water volume}=20\,\text{cm}\times10\,\text{cm}\times6\,\text{cm}
=1200 cm3=1.2 L=1200\,\text{cm}^3=1.2\,\text{L}

Work backwards to a missing dimension

When volume is known, reverse the multiplication. If a base area is known, divide volume by that area to find height. If two base edges are known, multiply them to get the base area first. For a face perpendicular to a known length, divide volume by that length to find the face area.

Inverse volume lab: find the missing dimension

Predict first: A cube has volume 216 cm³. What should you use to find its edge?

After trying the prediction, compare your method with this known-base example:

Known base edges

base area=4 cm×3 cm=12 cm2\text{base area}=4\,\text{cm}\times3\,\text{cm}=12\,\text{cm}^2
height=72 cm3÷12 cm2=6 cm\text{height}=72\,\text{cm}^3\div12\,\text{cm}^2=6\,\text{cm}

Roots belong to real dimensions

A square face uses a square root because its area is side × side. A cube uses a cube root because its volume is edge × edge × edge. These roots are shortcuts for reversing repeated multiplication; they are not a reason to apply a cube root to every cuboid.

Recognise the root

Square side=81 cm2=9 cm\text{Square side}=\sqrt{81\,\text{cm}^2}=9\,\text{cm}
Cube edge=216 cm33=6 cm\text{Cube edge}=\sqrt[3]{216\,\text{cm}^3}=6\,\text{cm}
(9 cm)2=81 cm2,(6 cm)3=216 cm3(9\,\text{cm})^2=81\,\text{cm}^2,\quad(6\,\text{cm})^3=216\,\text{cm}^3

Three graduated practices

Practice 1 — base area times height

A cuboid has a base 12 cm by 4 cm and volume 288 cm³. Find its height.

Show the solution

Reveal the check

base area=12 cm×4 cm=48 cm2\text{base area}=12\,\text{cm}\times4\,\text{cm}=48\,\text{cm}^2
height=288 cm3÷48 cm2=6 cm\text{height}=288\,\text{cm}^3\div48\,\text{cm}^2=6\,\text{cm}
48 cm2×6 cm=288 cm348\,\text{cm}^2\times6\,\text{cm}=288\,\text{cm}^3

Practice 2 — find a perpendicular face

A cuboid has volume 420 cm³ and a perpendicular length of 14 cm. What is the area of the matching face?

Show the solution

Reveal the check

face area=420 cm3÷14 cm=30 cm2\text{face area}=420\,\text{cm}^3\div14\,\text{cm}=30\,\text{cm}^2
30 cm2×14 cm=420 cm330\,\text{cm}^2\times14\,\text{cm}=420\,\text{cm}^3

Practice 3 — a square root in context

A square tile has area 144 cm². Find the side length of the tile.

Show the solution

Reveal the check

side=144 cm2=12 cm\text{side}=\sqrt{144\,\text{cm}^2}=12\,\text{cm}
12 cm×12 cm=144 cm212\,\text{cm}\times12\,\text{cm}=144\,\text{cm}^2

Watch out — easily mixed up

Quick recap

🎯 Mastery check

Answer all 6 — your progress is saved on this device.

  1. A cube has volume 125 cm³. What is its edge length?

  2. What is the volume of a cuboid 9 cm by 4 cm by 6 cm?

  3. A cuboid has base area 42 cm² and height 8 cm. What is its volume?

  4. A cuboid has volume 240 cm³ and height 10 cm. What is the area of its perpendicular face?

  5. A square face has area 81 cm². What is one side length?

  6. A cuboid has volume 504 cm³ and base area 56 cm². What is its height?