Volume with fractional edge lengths
Count fractional cubes and connect their count with the prism formula.
Before you begin
- Fraction multiplication
- Rectangular prism volume
Choose a cube that fits the dimensions
A rectangular prism can be packed with cubes whose side length is a suitable fraction of a unit. A cube with side 1/2 m has volume (1/2)³ = 1/8 m³, so eight such cubes fill one cubic metre. Count how many cubes fit along each edge, multiply those counts, then multiply by the volume of one cube. The count alone is not the volume in cubic metres.
Use base area times height
The same reasoning gives V = lwh even when lengths are fractions. The base area may be fractional, and multiplying it by the height produces cubic units. Convert mixed numbers to improper fractions to make multiplication easier. Check the scale using a nearby whole-number box and remember that internal dimensions describe capacity, whereas external dimensions may include the material of a container.
Worked example
A prism measures 1½ m by 1 m by 1/2 m. Find its volume using half-metre cubes.
Show the worked solution
- Along the edges fit 3, 2 and 1 half-metre cubes, giving 3 × 2 × 1 = 6 cubes.
- Each cube has volume 1/8 m³, so the total is 6/8 = 3/4 m³.
- Check with V = lwh: (3/2) × 1 × (1/2) = 3/4 m³.
3/4 m³.
Try it yourself
How many cubes of edge 1/2 cm fill a 1 cm cube?
Common mistakes
- A half-unit cube has volume one eighth of a cubic unit, not one half.
- Multiply the cube count by the volume of each small cube.
What can you explain now?
Find the volume of a box with internal dimensions 2½ cm by 1½ cm by 2 cm.
Compare with the explanation
7.5 cm³.
(5/2) × (3/2) × 2 = 15/2 = 7.5. The result is cubic centimetres because three lengths are multiplied.
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Common Core mathematics standards ↗