Scale a shape by k and its lengths grow by k, its areas by k², its volumes by k³. One exponent per dimension — that single sentence solves the whole topic.
The problem
Two similar solids have sides in the ratio 2 : 3. The smaller has volume 24 cm³. What is the larger's volume?
Volumes scale by the CUBE of the ratio: 24 × (32)³ = 24 × 278 = 81. A multiplies by 32 (a length move) and B by (32)² (an area move) — one exponent per dimension, and volume has three.
Double a square's side and it holds 2 × 2 = four of the originals — each dimension doubles independently. A cube doubles into 2 × 2 × 2 = eight. Length is one factor of k, area two, volume three. Nothing to memorise once you see the tiling.
| Quantity | Scales by | At k = 32 |
|---|---|---|
| length, perimeter | k | × 32 |
| area, surface area | k² | × 94 |
| volume | k³ | × 278 |
V = 24 · 278 = 81 cm³
the worked answer
areas 9 : 16 → k = √916 = 3 : 4
root the area ratio to get the side ratio
then volumes = 3³ : 4³ = 27 : 64
side ratio first, then re-raise
Never jump area-ratio → volume-ratio directly. The side ratio k is the bridge every crossing goes through.
"A photo is enlarged 150%…", "a model car is built at 1 : 24…", "the map scale is 1 cm : 5 km…" — all similar-figure questions. Enlarging a photo 150% multiplies ink AREA by 1.5² = 2.25; a 1 : 24 model holds 124³ of the real car's volume. The words change; k, k², k³ does not.
Prices work the same way when they follow area. A 25 cm pizza costs ₱500; a 15 cm one at the same price per unit of area costs 500 × (1525)² = 500 × 925 = ₱180 — not 500 × 1525 = ₱300, which scales by length alone.
Every length of a solid is increased by 20%. By what percent do its surface area and its volume increase?
k = 1.2
100% + 20%
area × 1.2² = × 1.44
up 44%
volume × 1.2³ = × 1.728
up 72.8%
“Area up 40%, volume up 60%” doubles and triples the 20%. The factor is 1.2, and it is the factor that gets squared or cubed; subtract 1 only at the end.
A square of side 6 cm is enlarged so that its area doubles. Find the new side.
k² = 2, so k = √2
un-square the area factor
side = 6√2 cm ≈ 8.49 cm
not 12 cm
Different: the area factor is not a perfect square, so k is irrational. Doubling the side would make the area 4 times as big.
On a map, 1 cm stands for 0.5 km. A park covers 6 cm² of the map. Find its real area.
1 cm² stands for 0.5 × 0.5 = 0.25 km²
square the length scale
6 × 0.25 = 1.5 km²
not 6 × 0.5 = 3
Different: the scale is given as a length with units. A map area uses the scale squared, units included.
Two similar solids have volumes in the ratio 27 : 125. The smaller has surface area 36 cm². Find the surface area of the larger. Cube-root down to k, then square back up.
k = 3√27125 = 35
volume ratio → side ratio
areas in the ratio 3² : 5² = 9 : 25
side ratio → area ratio
36 × 259 = 100 cm²
smaller to larger
36 × 12527 ≈ 166.7 uses the volume factor on an area. Any time the given ratio and the asked quantity have different dimensions, go through k.
A cone-shaped glass, point down, is filled to two thirds of its height. What fraction of the glass holds water? The water is itself a cone, similar to the glass.
k = 23
height of water ÷ height of glass
volume fraction = (23)³ = 827
under a third, even at two thirds’ height
This only works with the point at the bottom. Upside down, the air at the top is the similar cone, and the water is 1 − (air fraction).
Expect a ratio given in one dimension and asked in another — areas to lengths, volumes to areas — or a story: a model, a photo enlargement, a pizza priced by area, a cone part-filled. Find k first, then raise it to the power the question needs. Ten-second check: lengths change least, volumes most.
The problem
Two similar triangles have sides in the ratio 2 : 5. The smaller has area 8. What is the larger's area?
Area scales by k²: 8 × (52)² = 8 × 254 = 50. A multiplies by 52 — the length move — and D cubes, a volume move on a flat shape. Two dimensions, two factors of k.
