Radius x Height: Unpacking the Mystery of Volume in Cubes

[squarkman]

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

 

[Pengwuino]

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.

 

[flatmaster]

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.

 

[HallsofIvy]

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.