Volume When the Edges Are Fractions

Volume is still length × width × height, whether the edges are whole numbers or not.

Filling with smaller cubes

A box with half-unit edges can be filled with cubes of side 1/2, and those can be counted.

How many fit in a unit cube

Eight cubes of side 1/2 fit inside one unit cube, so each is 1/8 of a cubic unit.

Or just multiply

1/2 × 1/2 × 1/2 = 1/8, which agrees with the count.

The formula does not change

Volume is still length × width × height when the edges are fractions. A box 1/2 by 1/3 by 1/4 has volume 1/24 of a cubic unit. Fractional edges change the arithmetic, not the geometry.

Multiplying three fractions

Multiply all the numerators and all the denominators: (1 × 1 × 1)/(2 × 3 × 4) = 1/24. Simplify at the end, or cancel common factors first to keep the numbers small. Three factors is no harder than two, just longer.

Mixed-number edges

Convert every mixed number to an improper fraction before multiplying. Multiplying the whole parts and the fraction parts separately misses most of the volume, for the same reason it fails for area — there are more pieces than you counted.

Estimate to catch errors

A box roughly half by a third by a quarter of a unit should have a volume well under a tenth of a cubic unit. The answer 1/24 fits. Estimating with fractions means rounding to friendly ones and accepting a rough bracket.