# geometry

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Why do you use 1/3 in volume formulas?

• geometry -

any solid that comes to a vertex or point has the formula
V = (1/3) (base area) x (height)

I you know Calculus you could take the region bounded by the x-axis, the y-axis and a line with x-intercept of h and y-intercept of r and rotate it about the x-axis

The result will be a cone with radius r and height h
the equation of that line is
y = (-r/h)x + r

the generated volume
= π[integral]y^2 dx
= π[integral] (r^2x^2/h^2 - 2r^2x/h + r^2) dx
= π [r^2x^3/(3h^2) - r^2x^2/h + r^2x] from 0 to h
= π(r^2h/3 - r^2h + r^2h - 0)
= π(r^2)(h)/3
= (1/3)(πr^2)(h) or (1/3)basearea x height

When I was still teaching, I had a cylinder and a cone, both with the same radius and height.

In class we would fill the cone with water and pour it into the cylinder, and do that until the cylinder was full.
We were able to fill and pour THREE times, showing that the volume of the cone was
(1/3) of the volume of the cylinder.

notice
volume of cylinder = πr^2h
volume of cone = (1/3)πr^2h

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