Volume is the amount of space a solid takes up, measured in cubic units. A cubic centimetre is a cube one centimetre along every edge. As with area, the formula is a shortcut for a count.
Layers
Take a box 4 units long, 3 units wide and 5 units tall. The bottom layer is a rectangle of unit cubes, 4 x 3 = 12 of them. The box holds 5 such layers, so it holds 12 x 5 = 60 cubes.
Volume of a rectangular prism
area of the base: 4 x 3 = 12 square units number of layers: 5 volume = 12 x 5 = 60 cubic units which is the same as 4 x 3 x 5
The layer picture is worth keeping, because it generalises. For any prism, the volume is the area of the base multiplied by the height, whatever shape the base is.
Volume, capacity and surface area
| Measurement | What it counts | Units |
|---|---|---|
| Volume | space inside the solid | cubic units |
| Capacity | how much liquid it holds, the same space described differently | litres, millilitres, cups |
| Surface area | the total area of all the outside faces | square units |
A box 2 by 3 by 4 has a volume of 24 cubic units and a surface area of 2 x (6 + 8 + 12) = 52 square units. The two numbers answer different questions: one is how much fits inside, the other is how much wrapping paper it takes.
Common mistakes
Using square units for volume
Volume is cubic. Writing 60 square centimetres for a box is a unit error, and graders treat it as a wrong answer.
Adding the three dimensions
4 + 3 + 5 is 12 and means nothing here. Volume multiplies all three.
Check yourself
A box is 6 by 2 by 7. Find its volume.
84 cubic units.
A cube has edges of 5 cm. Volume and surface area?
Volume 125 cubic centimetres. Surface area 6 x 25 = 150 square centimetres.
A prism has a base area of 18 square cm and a height of 4 cm.
72 cubic centimetres. Base area times height works for any prism.
A box has a volume of 48 cubic units and a base of 4 by 3. How tall is it?
48 / 12 = 4 units.
See also
Sources
- Volume as a count of unit cubes, as introduced in United States upper elementary mathematics. Examples on this page are our own.