Radius x Height: Unpacking the Mystery of Volume in Cubes

In summary, the volume of a cone is 1/3 pi R^2 h and the answer is in units cubed because when you multiply three length measurements together, the result is always in volume units, regardless of the individual lengths. This concept can also be used for error checking in calculations.
  • #1
squarkman
10
0
Hi,
The volume of a cone is 1/3 pi R^2 h.
Why then is the answer in units cubed. Certainly the h is contributing a third dimension but in general math you don't see this happening
X^2 * Z = X^3
but the below is true
X^2 * X = X^3

Can a radius be multiplied by a height to make the dimension increase?
What's going on here?
Thx
 
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  • #2
Of course you can multiply by a height to make the dimension increase. X^2 * X = X^3 means you are multiplying the same variable. You can replace with Z and your new value is not X^3, but ZX^2 which has units of length cubed. Think of a rectangle with width X, length Y, and height Z. Surely you multiply them all together and you get length-cubed dimensions. X, Y, and Z all represent different lengths, but they represent length none-the-less.
 
  • #3
In fact, this can be used for error checking. If you arrived at an answer that should be volume and you end up with units of length squared, you know you did something wrong.
 
  • #4
squarkman said:
Hi,
The volume of a cone is 1/3 pi R^2 h.
Why then is the answer in units cubed. Certainly the h is contributing a third dimension but in general math you don't see this happening
X^2 * Z = X^3
but the below is true
X^2 * X = X^3

Can a radius be multiplied by a height to make the dimension increase?
What's going on here?
Thx
You are not distinguishing between numbers and their units. (X m)2(Z m)= (X2Z) m3, just as (X m)(Y m)(Z m)= (XYZ) m3.

If you multiply three length measurements together, the result is alway in volume units whether the lengths themselves are the same or not.
 

What is the formula for calculating the volume of a cube?

The formula for volume of a cube is V = l x w x h, where l is the length, w is the width, and h is the height.

How do you measure the radius and height of a cube?

To measure the radius of a cube, you would measure the distance from the center of one side to the opposite side. To measure the height, you would measure the distance from the base to the top of the cube.

Why is the radius and height important in calculating the volume of a cube?

The radius and height are important because they are the two measurements needed to calculate the volume of a cube. Without these measurements, the volume cannot be accurately calculated.

How does the volume of a cube change as the radius and height are increased or decreased?

The volume of a cube is directly proportional to the radius and height. This means that as the radius and height increase, the volume also increases. Conversely, as the radius and height decrease, the volume decreases as well.

What is the practical application of understanding the volume of a cube?

Understanding the volume of a cube is important in many scientific fields such as engineering, architecture, and chemistry. It allows for accurate calculations in designing structures, determining chemical reactions, and many other applications where volume is a crucial factor.

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