Few questions about surface area and volume

In summary, the factor of (4/3) in the formula for the volume of a sphere represents the ratio between the volume of a cone with height equal to the radius of the sphere and the surface area of the sphere. This was proven by Archimedes and is a constant created by integration. As for the surface area of an equilateral triangle, the factor of sqrt(3)/4 represents the ratio between the area of the triangle and the square of its side length, and has a geometric significance in the triangle's geometry.
  • #1
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When calculating the volume of a sphere, what does (4/3) represent? Why is it (4/3) * pi * r^3 .. and not some other number/fraction?

I'm also curious about the surface area of equilateral triangle. Why is it sqrt(3)/4 * a^2 ... What does sqrt(3)/4 physically represent in the geometry?
 
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  • #2
Well, if you rewrite the volume for the ball as [tex]V=\frac{1}{3}*4\pi{r}^{3}[/tex], recognize that this can be further simplified as:
[tex]V=\frac{1}{3}*r*S[/tex] where S is the surface area of the sphere.

Thus, the volume of the ball is equal to the volume of a cone of height "r" and base area S.

This is the gist result of how Archimedes proved the formula.
 
  • #3
Thanks. Here's what I just came up with for a possible physical (?) representation..

since, pi is the same as (2*pi*r)/(2*r)

V = ((4) * (2*pi*r) * (r^3)) / ((3) * (2*r))

or

V = (8 * pi * r^4) / (6 * r)


V = (4 * circumference) / (6 * radius)


Would the numerator represent 4 dimensions? Seems weird.
 
  • #4
cowah22 said:
Thanks. Here's what I just came up with for a possible physical (?) representation..

since, pi is the same as (2*pi*r)/(2*r)

V = ((4) * (2*pi*r) * (r^3)) / ((3) * (2*r))

or

V = (8 * pi * r^4) / (6 * r)


V = (4 * circumference) / (6 * radius)


Would the numerator represent 4 dimensions? Seems weird.
Since you are working in 3 dimensions, I doubt that! And 2 pi r^4 is the circumference of what?
 
  • #5
I meant,

V = (4/3 * pi * r^3) = (8 * pi * r^4) / (6 * r)

Which could be considered a ratio between whatever (8 * pi * r^4) is .. and (6 * r) which is (3 * Diameter)

disregard this:
V = (4 * circumference) / (6 * radius)
cowah22 was my secondary ID.
 
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  • #6
Does that make sense?
 
  • #7
Yes, but then you re-wrote 8*pi*r4 as 4*circumference. So it must be circumference = 2*pi*r4.

I think you're reading too much into what's essentially a constant created by integration (r2 -> r3/3, and the 4 comes from the surface area of a sphere formula)
 
  • #8
Office_Shredder said:
I think you're reading too much into what's essentially a constant created by integration (r2 -> r3/3, and the 4 comes from the surface area of a sphere formula)
Probably. Does (8*pi*r^4), or (Volume * (3*Diameter)) even have any geometric meaning/significance? I just thought it was interesting to see a 4th dimension in a sphere volume equation.
 
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1. What is the difference between surface area and volume?

Surface area refers to the measurement of the total area that covers the exterior surface of an object. On the other hand, volume is the measurement of the space occupied by an object, which includes its length, width, and height. In simpler terms, surface area is a 2-dimensional measurement while volume is a 3-dimensional measurement.

2. How do you calculate the surface area of a three-dimensional object?

The surface area of a three-dimensional object can be calculated by adding the areas of all of its faces. For example, for a cube, you would calculate the area of each of the six square faces and add them together. Different shapes have different formulas for calculating surface area, so it is important to know the formula for the specific shape you are working with.

3. Can you provide an example of why understanding surface area and volume is important?

Understanding surface area and volume is important in many fields, including architecture, engineering, and chemistry. For example, architects need to understand the surface area of a building's exterior in order to accurately estimate the amount of materials needed for construction. In chemistry, knowledge of the volume of a container is crucial for measuring and mixing substances accurately.

4. How does changing the dimensions of an object affect its surface area and volume?

Changing the dimensions of an object can have a significant impact on its surface area and volume. As the dimensions of an object increase, its surface area also increases, while its volume increases at a faster rate. For example, doubling the dimensions of a cube will result in a surface area that is four times larger, but a volume that is eight times larger.

5. Is there a relationship between surface area and volume for different shapes?

Yes, there is a relationship between surface area and volume for different shapes. As a general rule, as the surface area of a shape increases, so does its volume. However, the rate of increase for volume may vary depending on the shape. For example, a sphere has a smaller surface area compared to a cube with the same volume, but as the size of the sphere increases, its volume increases at a faster rate than the cube.

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