Overview
Similar figures share shape, so corresponding lengths change by one common scale factor. Areas contain two length dimensions and volumes contain three; they therefore change by the square and cube of that factor. Harder problems reverse this process or combine it with a difference between measurements. We will build the scale from the information supplied and track which kind of measurement the question ultimately requests.
Recover a length factor from perimeter
Two similar triangles have perimeters 28 and 42, and the smaller area is 20. Every side is multiplied by the same factor, so the perimeter is multiplied by that factor too. The larger-to-smaller length scale is 42/28 = 3/2. Area scales by (3/2)² = 9/4, so the larger area is 20·9/4 = 45.
The tempting answer 30 applies the length factor directly to the area. To see why this fails, imagine scaling both the base and its perpendicular height: the area formula contains their product, so the scale enters twice. A ratio is not merely a number to apply wherever it appears. Label it as a length, area or volume ratio, and label its direction as larger-to-smaller or smaller-to-larger. Those two labels prevent most accidental inversions and dimension mismatches.
Use differences to find an unknown scale
Two similar rectangles each have length-to-width ratio 3 : 2. Write the larger sides as 3u and 2u and the smaller sides as 3v and 2v. Their perimeters are 10u and 10v. A perimeter difference of 20 gives u − v = 2. Their areas are 6u² and 6v², so an area difference of 192 gives u² − v² = 32.
Factor the difference of squares: (u − v)(u + v) = 32. Substituting the known difference 2 gives u + v = 16. The sum and difference yield u = 9 and v = 7. The larger length is therefore 3u = 27. The distractor 21 is the smaller length, illustrating why identifying a scale parameter does not finish the interpretation. Both difference conditions are needed; the shared shape ratio alone cannot determine either rectangle’s size.
Take a cube root before using a geometric relation
Similar cones have smaller-to-larger volumes in ratio 8 : 27. The corresponding length ratio is the cube root, 2 : 3. If the larger cone is four centimeters taller, write the heights as 2u and 3u. Their difference is u = 4, so the larger height is 12.
The larger slant height is 13, and radius, vertical height and slant height form a right triangle. Thus its radius satisfies r² + 12² = 13², giving r² = 25 and positive radius r = 5. The answer 12 stops at the height; 8 reports the smaller height. This problem combines a scale law with a second geometric relationship. Keep those stages distinct: the volume ratio determines a linear relationship, the height difference fixes the size, and the right triangle finally determines the requested radius.
Apply the same scale within a divided triangle
In triangle ABC, DE is parallel to BC, with D on AB and E on AC. The smaller triangle ADE is similar to ABC, so AD/AB = DE/BC. With AD = 6, AB = 10 and DE = 9, the equation 6/10 = 9/BC gives BC = 15. Since BC belongs to the larger triangle, its length should exceed 9; the reversed-scale answer 27/5 fails that check.
For transfer, mark corresponding measurements before writing a proportion. Decide whether the given relationship is linear, squared or cubed, and convert it to the scale needed by the target. When differences are given, introduce common shape units rather than guessing the actual dimensions. After solving, check both the direction of change and the dimension of the result. A larger similar shape has larger corresponding lengths, but its area and volume grow faster; using that expectation helps detect a correct-looking fraction attached to the wrong kind of quantity.
Apply the method to a new case
Two similar containers have volumes 27 and 64 cubic centimeters. Their smaller surface area is 18 square centimeters. Find the larger surface area. The volume ratio is 27/64, so the smaller-to-larger length ratio is 3/4. Surface area uses the square of the length ratio, giving 9/16. Thus the larger area is 18·16/9 = 32 square centimeters. Multiplying 18 by 64/27 incorrectly applies a volume ratio to an area. Check that scaling length by 4/3 scales volume by 64/27 and area by 16/9, separately. Similarity is needed: two unrelated containers with those volumes need not have a fixed surface-area relationship. The units identify which power of the scale factor belongs to each measurement and prevent a plausible but dimensionally wrong multiplier.