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Study Notes 026, Solid Geometry 4:
By William E. Steinman:
The volume (V) of a cylinder is the product of its base area (B) and its altitude (h).
That is, V = Bh
This one seems to be intuitively obvious.
But, let us look at a cylinder to see it.
We know the area of a circle, the base, is πr2
So, by substitution, the equation becomes V = πr2h
Given a tomato can where the radius is 1.5 inches and the height is 4 inches, we can substitute directly to find to volume of tomatoes this can will hold.
V = 3.14 x 1.52 x 4 = 12.56 x 2.25 = 28.26 cubic inches.
That’s enough to flavor a small pot of goulash.
The volume of a cone (V) is 1/3 the product of the base area (B) and the altitude (h).
That is V = 1/3Bh
This it true of any cone, but let’s make it easy on ourselves by looking at the special case volume of a right circular cone.
This is given as V = 1/3πr2h
Again, we need calculus to prove this rigorously, but lets just look at it.
Take our tomato can from the previous problem. Now reduce one end of the can to a point. Instead of a can we have a cone of the same height.
Something like this.
If you put this cone upside down inside the original can you will see it takes up much less than half of the volume.
If you want to prove it is just 1/3 of the volume, you could fill the cone with sand and dump it into the can. You would find you would need three cones full of sand to fill the can. Try it if you wish.
So given a right circular cone with a base radius of 1.5 inches and a height of 4 inches we can substitute directly to find the volume of sand the cone would contain.
V = 1/3 x 3.14 x 1.52 x 4 = 9.42
Notice, that this is 1/3 of the volume we found for the can. Look at the equations and you can see it follows.
The volume of a sphere is given as V = 4/3πr3
I will make no attempt to prove this. Just take it on faith until we get to calculus.
Solving the equation, once we have it, is simple.
Given a sphere of radius 4 inches we find V = 4/3 x 3.14 x 43 = 267.95
The volumes of similar solids are to each other as the cube of their corresponding linear dimensions. Think about it.
This concludes the study of solid geometry. Next, I will take up trigonometry.
Did I hear someone say, “Oh crap”?