Similar figures keep the same shape while corresponding lengths change by one common factor. The difficulty is often choosing which factor belongs to the measurement. A factor that correctly changes a length will usually give the wrong area or volume if used unchanged.
Write the direction before calculating. In “small to large”, k means a large corresponding length divided by the small one. In the reverse direction the factor is 1/k. All three-dimensional worked models here are mathematical cuboids, including cubes; their volumes are derived from length × width × height.
Match the measurement to the factor
| Measurement | Small-to-large multiplier | Reverse multiplier | Typical units |
|---|---|---|---|
| Length or perimeter | k | 1/k | cm |
| Corresponding area | k² | 1/k² | cm² |
| Volume of a similar cuboid | k³ | 1/k³ | cm³ |
Why do the powers change? For a rectangle with dimensions a and b, the scaled area is (ka)(kb) = k²ab. A triangle has area ½bh; scaling its base and perpendicular height gives ½(kb)(kh) = k²(½bh). A polygon can be split into corresponding triangles, so their scaled areas add with the same factor.
For a cuboid with dimensions l, w and h, the scaled volume is (kl)(kw)(kh) = k³lwh. This is a direct calculation from the cuboid volume formula. Each corresponding rectangular face scales by k², so adding all six faces gives the same k² factor for total surface area.
Case 1: start with corresponding lengths
Two cuboids are geometrically similar. The smaller has dimensions 4 cm, 5 cm and 8 cm. The corresponding first dimension of the larger is 6 cm. Find the other dimensions, the volumes and the total surface areas.
From small to large, k = 6/4 = 3/2. Therefore the remaining dimensions are 5 × 3/2 = 7.5 cm and 8 × 3/2 = 12 cm. Check all three ratios: 6/4 = 7.5/5 = 12/8 = 1.5.
The small volume is 4 × 5 × 8 = 160 cm³. The volume multiplier is (3/2)³ = 27/8, giving 160 × 27/8 = 540 cm³. Direct multiplication of the larger dimensions confirms 6 × 7.5 × 12 = 540 cm³.
The small total surface area is 2(4 × 5 + 5 × 8 + 4 × 8) = 2(20 + 40 + 32) = 184 cm². The area multiplier is (3/2)² = 9/4, so the large total surface area is 184 × 9/4 = 414 cm². Directly, 2(45 + 90 + 72) also gives 414 cm².
| Quantity | Small cuboid | Large cuboid | Large ÷ small |
|---|---|---|---|
| First dimension | 4 cm | 6 cm | 3/2 |
| Second dimension | 5 cm | 7.5 cm | 3/2 |
| Third dimension | 8 cm | 12 cm | 3/2 |
| Total surface area | 184 cm² | 414 cm² | 9/4 |
| Volume | 160 cm³ | 540 cm³ | 27/8 |
A frequent wrong answer is 240 cm³, obtained by multiplying 160 by 1.5. That changes only one dimension’s contribution. The three corresponding dimensions each grow, so the multiplier must include three factors of 1.5.
Case 2: recover length scale from area
Two similar right isosceles triangles have areas 98 cm² and 162 cm². A leg of the smaller is 14 cm. Find the corresponding leg of the larger. Because similarity is given, we may recover a common length factor from the areas.
The small-to-large area ratio is 98:162 = 49:81. Therefore the corresponding length ratio is √49:√81 = 7:9. Equivalently, k² = 81/49, so the positive factor k = 9/7. The required leg is 14 × 9/7 = 18 cm.
Check the geometry rather than trusting the ratio alone: ½ × 14 × 14 = 98 and ½ × 18 × 18 = 162. Taking the positive root is appropriate because k here compares positive lengths. Perimeters also scale by 9/7 because every corresponding side, including the hypotenuse, has that factor.
Case 3: recover length and area from volume
Two cubes have volumes 216 cm³ and 1000 cm³. Find their edge-length ratio and total-surface-area ratio. A cube is a cuboid whose three dimensions are equal, so the cubed volume relationship applies directly.
Reduce the volume ratio: 216:1000 = 27:125. Take cube roots to get the length ratio 3:5. Hence the area ratio is 3²:5² = 9:25. It is not 27:125.
Independent dimensions provide a check: ∛216 = 6 cm and ∛1000 = 10 cm. The surface areas are 6 × 6² = 216 cm² and 6 × 10² = 600 cm². Reducing 216:600 gives 9:25. The small cube happens to have the same numerical value, 216, for volume and area, but the units and meanings remain different.
A larger area does not establish similarity
Compare rectangles measuring 8 cm by 2 cm and 8 cm by 4 cm. Their areas are 16 cm² and 32 cm², but they are not similar. Matching the long sides gives factor 8/8 = 1, while matching the short sides gives 4/2 = 2. There is no single common factor.
Do not take √2 and apply it to a side merely because the second rectangle has twice the area. The square-root method assumes similarity; it does not prove it. Similarly, a height measurement alone says nothing about the volume ratio unless the other dimensions are known to scale consistently.
Three solved practice tasks
- Two similar triangles have corresponding lengths in the ratio 3:5. The smaller area is 45 cm². Find the larger area. The small-to-large factor is 5/3, so the area is 45 × (5/3)² = 45 × 25/9 = 125 cm². The area ratio 45:125 reduces to 9:25.
- Similar cuboids have small-to-large volume ratio 27:125. A corresponding smaller edge is 12 cm. Find the larger edge and area ratio. The linear ratio is 3:5, giving the larger edge 12 × 5/3 = 20 cm. The corresponding area ratio is 9:25.
- A cuboid is reduced so each length becomes 0.4 of its original value. Its original volume is 375 cm³. Find the new volume. The multiplier is 0.4³ = 0.064. New volume = 375 × 0.064 = 24 cm³. Since 0.4 < 1, both the new dimensions and volume are smaller.
Keep units attached, distinguish total surface area from one face’s area, and state whether a ratio runs from the small shape to the large or the reverse. These checks prevent the correct power being applied in the wrong direction.
Frequently asked questions
What is the first step in a scale-factor question?
Name the direction, such as small to large, and match corresponding lengths. Dividing a target length by its matching source length gives the linear factor k.
Why is the area factor squared?
Area involves two length dimensions. Multiplying both by k multiplies their product by k². A triangle’s constant factor of one-half does not change that scaling.
Why is the cuboid volume factor cubed?
A cuboid’s length, width and height each multiply by k. Their product therefore multiplies by k × k × k = k³.
Can I use an area ratio as a length ratio?
No. Take the positive square root of the area factor to obtain the linear factor, provided the figures are stated to be similar and the areas correspond.
Does twice the volume mean twice the height?
For similar cuboids, the height factor is the cube root of 2, about 1.260. Doubling every length would multiply the volume by 8.
Do equal areas prove that two shapes are similar?
No. Area is only one measurement. Similarity also requires matching shape and consistent corresponding lengths; rectangles can have equal areas but different proportions.
Can these calculations give the mass or capacity of a real container?
Not from shape data alone. Capacity concerns internal dimensions, and mass also depends on material distribution and density. The examples here use mathematical cuboids and their geometric volumes.
Further reading
For the underlying mathematics, see AMSI’s length-similarity and area-scaling explanation. The mathematics topic-test collection is a separate resource page that currently shows a member login; specific files and their answers are not verified here.
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