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**Sphere volume, derivation**

To derive the formula for the volume of a sphere, a body can be created according to an idea of Galileo Galilei, which has the same cross-section as a hemisphere at the same heights. Such a body is created if a circular - homework for you - cone with the same base area and the same height is cut out of a circular cylinder with the base area radius r and the height r.

To derive the formula for the volume of a sphere, a body can be created according to an idea of Galilei, which has the same cross-section as a hemisphere at the same heights. Such a body is created if a circular cone with the same base area - https://domyhomework.club/math-homework/ - and the same height is cut out of a circular cylinder with the base area radius r and the height r.

If this residual body and a hemisphere with the same radius r are cut through a plane at the same height x, the cut surfaces have the same area - statistic homework help , as the following consideration shows on a cut through both bodies.

Since the highlighted triangle is isosceles, the following applies:

r1=x

For the area A1 of the circular ring we get:

A1=π r^2-π r1^2=π r^2-π x^2=π (r^2-x^2).

According to the Pythagorean theorem, for the hemisphere

r^2=x^2+r2^2 and therefore A2=π r2^2=π (r^2-x^2); therefore A1=A2.

According to Cavalieri's theorem the following then applies

Vhemisphere=Vcylinder-Vcone =π r^2h-1/3π r^2h=2/3π r^2h

and consequently the following applies to the volume of a sphere:

VSphere=4/3π r^2h

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