The Surface Area Ratio Calculator calculates how the surface areas and volumes of two similar solids compare, based on their length scale factor.
If every length of a solid is multiplied by k, its surface area is multiplied by k² and its volume by k³. Start from whatever you know — a scale factor, a surface area ratio, two surface areas or a volume ratio — and the calculator finds the other two.
Surface Area Ratio Calculator
Ratios of similar solids: surface area ratio = k², volume ratio = k³ — from a scale factor, an area ratio, two areas or a volume ratio.
Formula
Surface area ratio = k²
Volume ratio = k³
Working backward:
- k = √(surface area ratio)
- k = ∛(volume ratio)
- Volume ratio = (surface area ratio)^1.5
- Surface area ratio = (volume ratio)^(2/3)
Where k = length of solid 2 ÷ matching length of solid 1 (the linear scale factor).
How the Calculation Works
Step 1: Find the length scale factor k, either directly or from the ratio you entered.
Step 2: Square k to get the surface area ratio, because area has two dimensions.
Step 3: Cube k to get the volume ratio, because volume has three dimensions.
Variables Explained
| Variable | Meaning | Example |
|---|---|---|
| k | Length scale factor | 2 : 1 |
| SA₂ ÷ SA₁ | Surface area ratio | 4 : 1 |
| V₂ ÷ V₁ | Volume ratio | 8 : 1 |
Example Calculation
From a scale factor
Given: k = 2 (solid 2 is twice as long in every direction)
Result: surface area ratio = 2² = 4 : 1; volume ratio = 2³ = 8 : 1
From two surface areas
Given: SA₁ = 54 cm², SA₂ = 216 cm²
Area ratio = 216 ÷ 54 = 4; k = √4 = 2
Result: volume ratio = 2³ = 8 : 1
From a volume ratio
Given: volume ratio = 27 : 8
k = ∛(27 ÷ 8) = 1.5
Result: surface area ratio = 1.5² = 2.25 : 1 (9 : 4)
Scale Factor Reference Table
| Scale factor k | Surface area ratio | Volume ratio |
|---|---|---|
| 1.5 | 2.25 | 3.375 |
| 2 | 4 | 8 |
| 3 | 9 | 27 |
| 4 | 16 | 64 |
| 0.5 | 0.25 | 0.125 |
Shape Properties
What makes solids similar
Two solids are similar when they have the same shape and every length is in the same ratio — all angles equal, all matching edges scaled by the same k. Two cubes are always similar; two cylinders are only similar if their height-to-radius ratios match.
Dimensions Explained
The scale factor must compare matching lengths: radius to radius, height to height. A ratio written as 3 : 2 is the same as k = 1.5.
Frequently Asked Questions
What is the ratio of the surface areas of similar solids?
The square of the scale factor: SA₂ ÷ SA₁ = k².
How do you find the scale factor from surface area?
Take the square root of the surface area ratio: k = √(SA₂ ÷ SA₁).
How do you find the volume ratio from the surface area ratio?
Find k = √(area ratio), then cube it. An area ratio of 9 : 4 gives k = 3 : 2 and a volume ratio of 27 : 8.
If the scale factor is 3, what happens to surface area?
It becomes 9 times larger, and the volume becomes 27 times larger.
Do these ratios work for any two solids?
No, only for similar solids — the same shape at different sizes.
