Similarity Ratio Calculator
Enter the scale factor (ratio of corresponding side lengths) between two similar figures to find the resulting area and volume ratios.
Last updated: September 5, 2026
Frequently asked questions
If two similar shapes have sides in a 1:3 ratio, what's their area ratio?
1:9 - area scales with the square of the linear scale factor, so a side ratio of 3 gives an area ratio of 3² = 9.
If two similar solids have sides in a 1:3 ratio, what's their volume ratio?
1:27 - volume scales with the cube of the linear scale factor, so a side ratio of 3 gives a volume ratio of 3³ = 27.
Why does area scale by the square and volume by the cube?
Area is a two-dimensional measurement (length × length), so both dimensions scale by k, multiplying the area by k×k = k². Volume is three-dimensional (length × length × length), so all three dimensions scale by k, multiplying volume by k×k×k = k³.
Does this work for any similar shapes, or just squares and cubes?
Any similar shapes - triangles, circles, irregular polygons, spheres, or complex solids. As long as one figure is a uniform scaled copy of the other, the k² area rule and k³ volume rule apply regardless of the specific shape.
What's the area and volume ratio for a scale factor of 1?
Both ratios equal 1 (1:1) - a scale factor of 1 means the two figures are congruent (identical size), so there's no change in area (1²=1) or volume (1³=1).
If a model is built at 1:10 scale, how much less material does it need by volume?
1/1000 as much - volume scales as k³, so a linear scale factor of 1/10 gives a volume ratio of (1/10)³ = 1/1000 compared to the full-size original.
Does 'scale factor' always mean the shapes are similar, or could they be different shapes entirely?
Scale factor specifically applies to similar figures - shapes with identical proportions but different sizes (like two same-angle triangles). If the figures aren't similar (different shape, not just different size), the k² and k³ rules don't apply.