The volume of the sphere

Archimedes was able to demonstrate that the volume of the sphere is equal to two-thirds of the volume of the circumscribed cylinder.

This animation shows the cylinder (red, on the left), the sphere (blue, on the right) and a double cone, inscribed in the cylinder (green, in the center). The three solids are cut in half by a vertical plane (which contains the center of the sphere and the axes of the cylinder and the cone), in order to better see what happens. We want to show that the volume of the sphere is equal to the difference between the volume of the cylinder and that of the double cone.

With a click & drag on the figure you can change the point of view.

The three solids are also cut by a horizontal plane (that is, perpendicular to the axes of the double cone and the cylinder); here above we can see their sections, which are generally three circles (of which we see only three semicircles here). The height of this plane can be modified using the cursor under the figure and thus observe how the dimensions of the three circles vary: the red circle, section of the cylinder, always remains the same dimensions; the green circle, section of the cone, increases as the cursor is moved to the right and decreases in the other direction, reducing to a point when the cursor is completely to the left; the blue circle, a section of the sphere, increases as the cursor is moved to the left and decreases in the other direction, reducing to a point when the cursor is completely to the right.

On the right you see a right triangle in which:

• the hypotenuse (in red) is the radius of the sphere, and is therefore also equal to the radius of the red circle which is the section of the cylinder;
• a cathetus (in green) is the distance between the secant plane and the center of the sphere and is equal to the radius of the green circle, the section of the double cone;
• the other cathetus (in blue) is the radius of the blue circle which is the section of the sphere.

By the Pythagorean Theorem (applied to circles instead of squares) we can conclude that the area of ​​the red circle is the sum of the area of ​​the green circle and the area of ​​the blue circle.

By Cavalieri’s principle, this guarantees that the volume of the cylinder is the sum of the volume of the double cone (which is 1/3 of the volume of the cylinder) and that of the sphere: that is, the volume of the sphere is 2/3 of the cylinder volume.

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