# SAT Similar Shapes: Length, Area and Volume Ratios | topin

On this page

## Establish similarity before applying a power rule

College Board’s [Geometry and Trigonometry content](https://satsuite.collegeboard.org/sat/whats-on-the-test/math/types/geometry-trigonometry) includes area, volume and triangles. The mathematical scaling rules come from the formulas: area multiplies two lengths, while volume multiplies three. A scale factor applies to corresponding dimensions. Do not use a similarity shortcut when the question changes only one dimension or does not establish that the shapes are similar.

Our starting labels are original, new and corresponding. Identify a pair of matching lengths, then form new divided by original to obtain k. Keep the direction explicit. If an old side is 4 and its matching new side is 6, k = 6/4 = 3/2\. The reverse comparison has factor 2/3\. Both are mathematically valid, but only one matches the requested direction.

For triangles, match vertices by the angle or correspondence information provided. Do not pair sides merely because they occupy a similar place on your rough sketch. A rotated or reflected triangle can preserve correspondence while looking different. Write the matching labels before forming ratios. The diagrams you draw are aids to the stated relationships, not additional evidence that an unstated similarity condition is true.

## Original example: obtain an area ratio from a side ratio

Two similar triangles have corresponding sides 8 centimetres and 12 centimetres. The smaller triangle’s area is 24 square centimetres. The length scale from smaller to larger is k = 12/8 = 3/2\. The area scale is (3/2)² = 9/4, so the larger area is 24 × 9/4 = 54 square centimetres. Doubling 24 or multiplying it by 3/2 would use the wrong factor.

You can verify through a base-and-height example. Suppose the smaller triangle has base 8 and height 6, giving area 8 × 6 / 2 = 24\. Under scale 3/2, the larger base is 12 and the larger height 9, giving 12 × 9 / 2 = 54\. Both lengths scale, which explains the square. The factor is not an arbitrary rule unrelated to the formula.

If instead the triangles merely share base 8 and one has a different height, the area ratio follows the height ratio alone. That is a different condition. Use the information actually given rather than recognising two triangles and assuming similarity. A correct area formula can still produce the wrong answer when the relationship between the shapes is incorrectly modelled.

## Original example: recover a length ratio from areas

Two similar rectangles have areas 45 and 80 square units. What is the ratio of a corresponding length in the smaller rectangle to the larger? The area ratio is 45/80 = 9/16\. The length ratio is the positive square root, 3/4\. The positive root is appropriate because side lengths are positive. Reporting 9/16 gives an area ratio when a length ratio is requested.

If the larger rectangle’s corresponding side is 20, the smaller side is 20 × 3/4 = 15\. Check through a compatible example: the smaller rectangle could be 15 by 3, with area 45; the larger would be 20 by 4, with area 80\. Both dimensions have the same length factor 4/3 from smaller to larger. The original question did not require these particular dimensions; they simply verify the ratio.

When the question asks the percentage increase in length, the area percentage is not interchangeable. The larger length factor is 4/3, giving a 33 1/3% increase, while the area factor is 16/9, giving a different increase. Identify whether the target is a length, area or percentage change before choosing the final expression. Link the units to the exponent so the relationship stays visible.

## Original example: volumes of similar solids

Two similar solids have volumes 64 and 216 cubic units. Their volume ratio is 64/216 = 8/27\. The corresponding smaller-to-larger length ratio is the positive cube root, 2/3\. The corresponding area ratio is then (2/3)² = 4/9\. One given volume ratio can therefore produce both a length ratio and an area ratio, but each needs the appropriate power.

A cube example verifies the arithmetic: sides 4 and 6 give volumes 4³ = 64 and 6³ = 216, while face areas are 16 and 36, ratio 4/9\. Similarity means every corresponding dimension scales by the same amount, not just one selected edge. The method also applies to other similar solids because the volume formula’s dimensions collectively contribute the cube of the length factor.

If a question gives a volume multiplier of 125, the length multiplier is 5 and the area multiplier 25\. If it gives a length multiplier of 125, the volume multiplier is 125³. The same number in different roles leads to radically different results. Our recommended annotation writes the known measure above the ratio and the requested measure beside the final answer before any power or root is applied.

## Original example: one changed dimension is not full similarity

A cylinder has radius r and height h, so volume is πr²h. If the radius doubles and height remains h, the new volume is π(2r)²h = 4πr²h, four times the original. If both radius and height double, the new volume is π(2r)²(2h) = 8πr²h. The eightfold similarity result requires both changes, not merely the word cylinder.

If the radius triples and height is divided by nine, the new volume is π(3r)²(h/9) = πr²h, unchanged. Here the dimension multipliers combine as 3² × 1/9 = 1\. This is a formula-substitution problem rather than a similar-solid shortcut, because radius and height have different factors. Write the changed dimensions into the formula to avoid an inappropriate cube rule.

For a rectangle whose length doubles while width halves, area also stays unchanged: (2L)(W/2) = LW. The shape is generally not similar to the original unless further special conditions apply. An unchanged area does not imply unchanged dimensions or similarity. These original examples help distinguish a geometric relationship from a numerical coincidence in one measure.

## Original example: percentage changes in dimensions

A circular sign’s radius increases by 20%. Its radius multiplier is 1.20, so area multiplies by 1.20² = 1.44\. Area increases by 44%, not 40% and not 20%. Circumference, which uses one length through 2πr, increases by 20%. A statement about the same physical shape can therefore involve different percentage changes depending on the requested measure.

For a similar solid whose lengths decrease by 10%, k = 0.90\. Volume becomes 0.90³ = 0.729 of the original, a decrease of 27.1%. Area becomes 0.81 of the original, a decrease of 19%. Do not multiply the original percentage by two or three as an exact rule; the powers contain the effect of the changed dimensions. Use the [percentage guide](/articles/sat-reverse-percentages) when recovering an original amount.

If area increases by 21% for similar shapes, the area multiplier is 1.21, so k = √1.21 = 1.10 and lengths increase by 10%. Reverse questions need a root rather than another square. Check forwards: 1.10² = 1.21\. This quick verification catches confusing 21% with a length multiplier of 1.21, which would produce a larger area change.

## Choose the exponent from the quantity, then verify units

__Original scale-factor decision table for similar figures__
| Given and requested             | Operation                           | Example                                     |
| ------------------------------- | ----------------------------------- | ------------------------------------------- |
| Length factor to area factor    | Square k                            | 3/2 becomes 9/4                             |
| Length factor to volume factor  | Cube k                              | 2 becomes 8                                 |
| Area ratio to length ratio      | Positive square root                | 9/16 becomes 3/4                            |
| Volume ratio to length ratio    | Positive cube root                  | 8/27 becomes 2/3                            |
| Volume ratio to area ratio      | Cube root, then square              | 8/27 becomes 4/9                            |
| Only selected dimensions change | Substitute factors into the formula | Radius ×2, height unchanged gives volume ×4 |

Use the [geometry overview](/sat/math/geometry-and-trigonometry) for the broader domain and the [word-problem units method](/articles/sat-math-word-problems-units) for conversion checks. Practise fresh questions that mix length, area and volume targets so the heading does not reveal the exponent. Label wrong attempts by correspondence, direction, similarity condition or exponent rather than putting every mistake under “geometry”.

[topin’s free SAT mock](/sat/practice-test), marked on the official scale, can check transfer to the timed exam. If you use a calculator, enter brackets and powers explicitly, but decide the relationship first. A correct calculation of k³ cannot rescue a case that changes only the radius. Finish with the quantity’s units and a quick forward check: the proposed length factor should recreate the stated area or volume ratio.

## Try a full SAT mock free

Timed like test day, every section scored, every answer explained.

[Take the free mock ](/sat/login)

## FAQs

Do similar shapes have the same area ratio as side ratio?

No. An area ratio is the square of the corresponding length ratio.

How do I recover side ratio from a volume ratio?

For similar solids, take the positive cube root of the volume ratio.

Does doubling a cylinder’s radius multiply volume by eight?

Only if height doubles too. With unchanged height, doubling radius multiplies volume by four.

What happens to area when radius increases by 20%?

The area factor is 1.20² = 1.44, an increase of 44%.

When should I avoid the similarity shortcut?

When corresponding dimensions do not share one scale factor or similarity has not been established. Substitute the actual changes into the formula.

## Sources (checked 5 October 2026)

* [College Board: Geometry and Trigonometry content (checked 5 October 2026)](https://satsuite.collegeboard.org/sat/whats-on-the-test/math/types/geometry-trigonometry)
* [College Board: Problem-Solving and Data Analysis content (checked 5 October 2026)](https://satsuite.collegeboard.org/sat/whats-on-the-test/math/types/problem-solving)
* [College Board: using the Student Question Bank (checked 5 October 2026)](https://satsuite.collegeboard.org/practice/student-question-bank)

## Related articles

[All SAT study guides ](/sat)

* [SATStudy plans & strategySAT Math timing: a 35-minute module strategyBuild a practical SAT Math pacing plan for 22 questions in 35 minutes, with checkpoints, skip decisions and an original example of choosing the fastest method.Updated 5 Oct 2026](/articles/sat-math-module-pacing)
* [SATStudy plans & strategySAT Math word problems: equations, rates and unitsTranslate SAT word problems into equations using quantities and units, with original ticket, rate, mixture and conversion examples and common modelling traps.Updated 5 Oct 2026](/articles/sat-math-word-problems-units)
* [SATStudy plans & strategySAT quadratic parameters: discriminants and root conditionsSolve SAT quadratic parameter questions using discriminants, vertex form and root relationships, with original one-root, no-root and non-quadratic examples.Updated 5 Oct 2026](/articles/sat-quadratic-discriminant-parameters)
* [SATStudy plans & strategySAT reverse percentages and successive changesSolve SAT reverse-percentage questions with multipliers, different base amounts, successive changes and percentage-point comparisons using original examples.Updated 5 Oct 2026](/articles/sat-reverse-percentages)