Chapter 14.3 — Exercise 14.2 — Volumes

Three Straightforward Volumes

A cube's volume formula is nothing more than the cuboid formula with b=h=l, exactly the same relationship the surface-area formulas followed in the earlier lessons of this chapter.

Multiplying three given dimensions directly answers the first set of problems, decimals included:

The decimal row isn't fundamentally different from the whole-number ones — 8.2 × 5.3 × 2.6 is exactly the same multiplication as any other three-number product, just with decimal points to track carefully through each step rather than whole numbers.

From Cubic Metres to Litres

Tank capacity problems add one extra step beyond a plain volume calculation: converting the result into litres, using the fact that 1 cubic metre holds exactly 1000 litres.

Every dimension here is given as metres-and-centimetres (3 m 20 cm, for instance) rather than a single decimal — the centimetre part has to be converted to a decimal fraction of a metre (20 cm = 0.20 m) before multiplying, exactly the same unit-matching discipline needed throughout this entire course whenever two different units show up in the same problem.

Halving an Edge Doesn't Halve the Volume

If a cube's edge is reduced to half its original length, what happens to its volume? Writing the original side as l and the new side as l/2:

The new volume is just one-eighth of the original — not one-half, even though the edge itself was only halved. The amount of volume lost is V₁ − V₁/8 = 7V₁/8, so the cube loses seven-eighths of its original volume while its edge loses only half its original length. This is a useful reminder that volume scales with the cube of a length change, not the length change itself — the same kind of scaling jump already seen with area scaling as the square of a length change back in the areas-of-plane-figures chapter.

Two More Cube Volumes, Straight From the Formula

Cubing a decimal side length follows the same process regardless of how many decimal places are involved:

Counting Bricks Needed for a Wall

A wall 8 m long, 6 m high, and 22.5 cm thick needs to be filled with bricks measuring 25 cm × 11.25 cm × 6 cm. Converting the wall's length and height into centimetres first (8 m = 800 cm, 6 m = 600 cm) keeps every dimension in the same unit before either volume is calculated:

This "divide the big volume by the small volume" approach assumes the bricks fit together perfectly with no wasted gaps — a reasonable assumption for a neatly built wall, and the same logic used in the remaining "how many small pieces fit in a big one" problems below.

Comparing a Cuboid Against a Cube of Similar Size

A cuboid measuring 25 cm × 15 cm × 8 cm is compared against a cube of edge 16 cm:

Even though the cube's edge (16 cm) sits comfortably between the cuboid's largest and smallest dimensions, the cube ends up considerably roomier — a further sign that a compact, cube-like shape tends to enclose more volume than a stretched-out cuboid built from similar-sized numbers.

The Volume Hidden Inside a Wooden Box's Walls

A closed wooden box, 1 cm thick, measures 5 cm × 4 cm × 7 cm on the outside. Finding how much wood was used means subtracting the hollow inner volume from the solid outer volume — and the inner dimensions shrink by 1 cm on each side, so 2 cm off every outer dimension in total:

The height shrinks by only 2 (7 to 5), not 4, since the box is only 1 cm thick — the wall thickness gets subtracted once from each side of every dimension, never twice from the same side.

Cutting Small Cubes and Small Cuboids From a Larger Block

Two related problems ask how many identical small solids can be cut from one larger one — both solved by dividing the big volume by the small one, exactly as the brick-wall problem did:

Solving for a Missing Height Instead of a Volume

A cuboidal vessel 30 cm long and 25 cm wide needs to hold 4.5 litres of water — what height does it need? This time the volume is known and a dimension is missing, so the formula gets solved in reverse, after converting litres to cubic centimetres (1 litre = 1000 cu. cm):

Exactly like the "find the side from a known surface area" problem in the previous lesson, whenever a volume is given and a dimension is missing, writing the formula down first and substituting every known value is what turns the problem into simple algebra rather than a guess.

Two Genuinely Different Questions About the Same Solids

Across this chapter, surface area and volume have answered two different questions about the exact same cubes and cuboids — how much material wraps around the outside, and how much space sits inside. Both concepts return in later mensuration chapters applied to cylinders, cones, and spheres, where curved surfaces replace flat rectangular ones, but the same underlying distinction — covering versus filling — stays exactly the same.