Volume of a Sphere

Materials

Volumes worksheet p. 2-3 (one per person)

Computer or other internet connected device (optional, one per person or pair or class)

Stack of coins/counters or paper (optional)

Cavalieri's Principle

If the cross-sectional areas of two solids are equal at every height h then their volumes are equal.

Cavalieri's principle is a three-dimensional analog to shearing two-dimensional objects and obtaining the same area. The same way that we can shear a vertex parallel to a base in a triangle and obtain a triangle with the same area (see: Shearing Triangles ). In a similar manner, we find that we can "shear" a three dimensional object and maintain the volume. For example, if we consider a right cylinder with height h and radius R, an oblique cylinder with the same height and radius will have the same volume. Just like the apex of a triangle needs to lie on the line parallel to the base to generate a new triangle with the same area, translating a base of a right cylinder in a plane parallel to the opposite base will produce an oblique cylinder with the same volume.

If we think of having a stack of thin discs, we can shift each one slightly to the side to form an oblique stack with the same volume. The oblique cylinder simply requires shifting wafer thin volumes parallel to the base from the top to the bottom so that the sides are smooth.

Note: We can use a ream of paper to illustrate the same concept with a rectangular prism.

Cavalieri's theorem takes this notion one step further and tells us that I can deform the cross-section of a solid at any height h into any shape and as long as the areas at the same height are the same, the volume remains the same as well. This allows us to find the volume of a hemisphere by connecting cross-sections to the cross-sectional areas outside an inverted cone inscribed in a cylinder.

Hemisphere, Cylinder, and Cone

To derive the volume formula for a sphere of radius R we will apply Cavalieri's principle to a hemisphere of radius R and the volume in a cylinder of radius and height R outside an inverted cone with radius and height R. Use the slider on the applet below to see the cross-sectional areas in these solids that we are going to show are the same at every height h.

Consider the cross-sections of the hemisphere and the cylinder outside the inverted cone.

At height h we can describe the cross-section of the hemisphere as a circle of radius r. The corresponding cross-section in the other solid is an annulus (i.e. a "washer") with an outer radius of R and inner radius equal to the radius of the cone at height h that we will call x for now. Because the inverted cone is a right cone with height and radius R and the part of the cone from the apex to height h is similar to the cone, the ratio of height h and radius x of the cross-section should also be equal to the ratio of height to radius of the cone.

By solving the proportion for x, we see that the inner radius of the annulus is equal to the height of the cross-section. Next, we express the areas of the cross sections.

Now we use the pythagorean theorem to relate the radius r of the circular cross-section of the hemisphere at height h to the radius R of the hemisphere to show that the cross-sectional area is equal to the area of the annulus at height h.

Because the cross-sectional areas of both solids are the same at every height h, according to Cavalieri's principle, the volumes of the solids are equal. That is, the volume of a hemisphere of radius R is equal to the volume of a cylinder of height and radius R with an inverted cone of height and radius R removed.

Volume Formula for a Sphere of Radius R

We showed above that the volume of a hemisphere is given by

Discussion

Note that one major difference between this proof and the explorations we did with the models and self-sticking sand used a cylinder and cone with radii R and heights 2R. In order to apply Cavalieri's principle, we needed to use a cone and cylinder that were the same height R as the hemisphere in order to obtain equal cross-sectional areas at every height h. Using the cylinder of height 2R could be compared to the entire sphere using Cavalieri's principles but only if a double-cone of height 2R and radius R is removed from the cylinder.

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