What is the relation between the volume and total surface area of cuboid
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This question is asking how each is altered after a similar change. An area has two dimensions. A volume has three. Explore the results of, say doubling their sizes. or, tripling. Oh, yeah. How do you define “doubling”?
With a few examples for both, a pattern should become evident.
Hint. Start with a square and a cube. The sizes might be anything. Perhaps thumb widths.
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Aravind, Computer science engineering student, Maybe a future genius
Answered Oct 15, 2016
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I'll try to use an analogy to explain based on what I feel is a reasonable way to understand.
I'm going to assume we are only using integers or whole numbers, because I will only be considering solids that are finite.
Consider one dimension. Graphically speaking, it's just the x axis and the points on it.Now the length of say a line gives the points contained in it. That is, every point contained within the end points. (Space in one dimension)Now consider two dimensions. Graphically speaking, it's the x and y axes and numbers on them. The regular one where the origin is at 0.Now the area of say a shape gives the points contained in it, That is, every point contained within the boundary. (Space in two dimensions)Now consider three dimensions. Graphically speaking, it's the x, y and z axes and numbers on them. The regular one where the origin is at 0.Now the volume of say a body gives the points contained in it. That is, the boundary of the body. (Space in three dimensions)
You may also notice that, when it comes to a two dimensional figure we may calculate length of a part of it. Similarly, we can find area of a part of a 3 dimensional figure. For example, total surface area of a cuboid, we may also find surface areas of just a few faces or even lengths of edges.
That is, area gives the amount of space contained in a body of two dimensional realm while volume gives the amount of space contained in a body existing in a three dimensional realm.
Now, a three dimensional space can be represented as three - one dimensional lines that generate three - two dimensional places. So we can calculate the space contained in a lower dimensional description of a part of a higher dimensional body as well.
Being conceived and evolved in a three dimensional world, it is rather difficult for us to understand and imagine 4 dimensions, hence the lack of a common word to explain the space contained in a 4d space.
However, I found some good stuff you might like.
Four-dimensional space - Wikipedia, the pictures are really trippy.
What is the 4D equivalent of 3D volume?, I'm not sure about the credibility of anything here but seems alright.
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Aladin Civic, Engineering student, dreams of working and living in Dubai
Answered Feb 9
Originally Answered: Is there a relation between area and volume?
"To put it short, the surface area should be the derivative or partial derivative of the volume.
This is because if the figure is magnified by some little factor, the incremental volume roughly equals the surface area times the incremental height.
However, due to the nature of derivatives, the magnification is one sided, which means some of the faces remain unscaled, and thats why when you take the derivatives of volume, you should get something smaller or equal to the surface area, leaving out some surfaces unscaled."
Credits to Trevor Cheung, who answered this to a similar question.
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Juvie Valerio, works at Philippine Long Distance Company
Answered Oct 20, 2016
Imagine a cube 1 foot on a side (substitute “meter” for “foot” if you like). Its volume will be 1 cubic foot. Each side is one square foot, so its total surface area is 6 square feet (because the cube has 6 sides).
Now imagine a cube 3 feet on a side. Its volume will be 27 cubic feet (3 × 3 × 3). Each side is 9 square feet (3 feet by 3 feet), so its total surface area is 54 square feet.
What has happened? The linear dimensions are 3 times bigger. The surface area has increased 9 times (54 ÷ 6). But the volume has increased 27 times (27 ÷ 1).
Let's calculate surface area to volume ratios:
For the 1-foot cube, 6:1.
For the 3-foot cube, 54:27, or 2:1.
For a 10-foot cube, 600:1000, or 0.6:1.
As the cube gets bigger, the volume increases much more rapidly than the surface area, because the volume increases as the cube of the linear dimension, but the surface area increases as the square. This relationship applies, not just to cubes, but to spheres and any other fixed shape.
With a few examples for both, a pattern should become evident.
Hint. Start with a square and a cube. The sizes might be anything. Perhaps thumb widths.
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Aravind, Computer science engineering student, Maybe a future genius
Answered Oct 15, 2016
Continue Reading
I'll try to use an analogy to explain based on what I feel is a reasonable way to understand.
I'm going to assume we are only using integers or whole numbers, because I will only be considering solids that are finite.
Consider one dimension. Graphically speaking, it's just the x axis and the points on it.Now the length of say a line gives the points contained in it. That is, every point contained within the end points. (Space in one dimension)Now consider two dimensions. Graphically speaking, it's the x and y axes and numbers on them. The regular one where the origin is at 0.Now the area of say a shape gives the points contained in it, That is, every point contained within the boundary. (Space in two dimensions)Now consider three dimensions. Graphically speaking, it's the x, y and z axes and numbers on them. The regular one where the origin is at 0.Now the volume of say a body gives the points contained in it. That is, the boundary of the body. (Space in three dimensions)
You may also notice that, when it comes to a two dimensional figure we may calculate length of a part of it. Similarly, we can find area of a part of a 3 dimensional figure. For example, total surface area of a cuboid, we may also find surface areas of just a few faces or even lengths of edges.
That is, area gives the amount of space contained in a body of two dimensional realm while volume gives the amount of space contained in a body existing in a three dimensional realm.
Now, a three dimensional space can be represented as three - one dimensional lines that generate three - two dimensional places. So we can calculate the space contained in a lower dimensional description of a part of a higher dimensional body as well.
Being conceived and evolved in a three dimensional world, it is rather difficult for us to understand and imagine 4 dimensions, hence the lack of a common word to explain the space contained in a 4d space.
However, I found some good stuff you might like.
Four-dimensional space - Wikipedia, the pictures are really trippy.
What is the 4D equivalent of 3D volume?, I'm not sure about the credibility of anything here but seems alright.
5.6k Views
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Comment...

Aladin Civic, Engineering student, dreams of working and living in Dubai
Answered Feb 9
Originally Answered: Is there a relation between area and volume?
"To put it short, the surface area should be the derivative or partial derivative of the volume.
This is because if the figure is magnified by some little factor, the incremental volume roughly equals the surface area times the incremental height.
However, due to the nature of derivatives, the magnification is one sided, which means some of the faces remain unscaled, and thats why when you take the derivatives of volume, you should get something smaller or equal to the surface area, leaving out some surfaces unscaled."
Credits to Trevor Cheung, who answered this to a similar question.
770 Views
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Juvie Valerio, works at Philippine Long Distance Company
Answered Oct 20, 2016
Imagine a cube 1 foot on a side (substitute “meter” for “foot” if you like). Its volume will be 1 cubic foot. Each side is one square foot, so its total surface area is 6 square feet (because the cube has 6 sides).
Now imagine a cube 3 feet on a side. Its volume will be 27 cubic feet (3 × 3 × 3). Each side is 9 square feet (3 feet by 3 feet), so its total surface area is 54 square feet.
What has happened? The linear dimensions are 3 times bigger. The surface area has increased 9 times (54 ÷ 6). But the volume has increased 27 times (27 ÷ 1).
Let's calculate surface area to volume ratios:
For the 1-foot cube, 6:1.
For the 3-foot cube, 54:27, or 2:1.
For a 10-foot cube, 600:1000, or 0.6:1.
As the cube gets bigger, the volume increases much more rapidly than the surface area, because the volume increases as the cube of the linear dimension, but the surface area increases as the square. This relationship applies, not just to cubes, but to spheres and any other fixed shape.
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