Why Don't Surface Area & Volume Ratios of Spheres Match?

In summary, the ratio of the surface area of spheres is not equal to the ratio of volume of spheres. This is because the surface area is proportional to the square of the radius, while the volume is proportional to the cube of the radius. This can be understood by comparing lines, squares, and cubes with different side lengths. Additionally, when considering two hot coals taken out of a fire, the smaller one will cool down faster due to its larger surface area to volume ratio.
  • #1
vikvon
1
0
Howcome the ratio of the surface area of spheres are not equal to the ratio of volume of spheres?

For example if I had two spheres, with an aspect ratio of 1:4 when comparing surface areas, yet I have 1:8 between those same spheres when comparing volume.

This doesn't make sense to me, shouldn't the ratio be the same?
 
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  • #2
the surface area is proportional to the square of the radius

the volume is proportional to the cube of the radius
 
  • #3
This might help...

A line of length 2cm consists of TWO 1cm lines.

A square with side length 2cm consists of FOUR squares with side length 1cm.

A cube with side length 2cm consists of EIGHT cubes with side length 1cm.

And imagine if you took two hot coals out of a fire, one small one and a big one, which one would cool down fastest, and why?
 

Related to Why Don't Surface Area & Volume Ratios of Spheres Match?

1. What is the difference between surface area and volume ratios of spheres?

Surface area and volume ratios of spheres refer to the relationship between the surface area and volume of a sphere. The surface area ratio is the measure of the amount of area on the surface of a sphere compared to its total volume, while the volume ratio is the measure of the amount of volume enclosed by the surface of a sphere compared to its total volume.

2. Why don't the surface area and volume ratios of spheres match?

The surface area and volume ratios of spheres do not match because the surface area increases in proportion to the square of the radius, while the volume increases in proportion to the cube of the radius. This means that as the radius of a sphere increases, the surface area increases at a slower rate than the volume, resulting in a difference between the two ratios.

3. How does the size of a sphere affect its surface area and volume ratios?

The size of a sphere directly affects its surface area and volume ratios. As the size of a sphere increases, the surface area ratio decreases while the volume ratio increases. This is due to the relationship between the radius, surface area, and volume of a sphere.

4. Can the surface area and volume ratios of spheres ever be equal?

No, the surface area and volume ratios of spheres can never be equal. As previously mentioned, the surface area and volume of a sphere increase at different rates, resulting in a difference between the two ratios. This is a fundamental property of spheres and cannot be changed.

5. What is the significance of understanding the surface area and volume ratios of spheres?

Understanding the surface area and volume ratios of spheres is important in various fields of science, such as physics, chemistry, and biology. It allows us to calculate and predict the behavior and properties of spheres, which are commonly found in nature and used in many man-made objects. Additionally, understanding these ratios can also provide insight into the structural integrity of spheres and their ability to withstand external forces.

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