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# how does surface area to volume ratio work?

I'm struggling with a question from my textbook.

Question:

A student stated the same shape scaled down should retain the same surface-area-to-volume ratio, the student’s reason being ‘the shapes stay the same’. Do you agree with this student? Explain your decision.

I think the answer is no the SA:V would change but i'm not sure and I don't know how to explain it.

Any help is appreciated, thank you :)

### 3 Answers

- frank lynnLv 64 weeks ago
So use two spheres and determine if the SA:V changes related to size. SA = 4(Pi)r^2, while V = 4(Pi)r^3. SA:V = 1/r. That means as the radius gets larger, the ratio gets smaller and the student is wrong.

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- VamanLv 74 weeks ago
square of 10 cm. have volume=10^3 Surface area

6*10^2. Ratio volume to area= 10^3/6 10^2= 10/6=5/3

Now take square of 2cm. Volume= 2^3=8. Surface area 6*4= 24, The ratio = 1/3. Now scale down values\

5/3*3=5. Sides were scaled down 5 times.

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- GodLv 74 weeks ago
consider the volume of a sphere = 4/3 Pi r^3

the surface area of a sphere = 4 Pi r^2

In general the volume increases with the cube of a number. The area increases with the square of a number. So the volume increases faster than the area as the size increases.

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