Cylinders, Cones, Spheres, & Volume

In this activity we will be deriving the volume formulas for cones and spheres based on what we know about cylinders. We will be making a sphere, cylinder, and cone with roughly the same radius and height. Then we will compare their volumes to determine the formulas.

We will begin by forming a sphere of Play-Doh. Try to get as close as possible to a perfect sphere. Now we will take some transparent material and form a snug cylinder around the sphere; use tape to keep the shape of the cylinder. Trim the height of the cylinder so it matches the height of the sphere.

Now we will flatten the Play-Doh sphere into the bottom of the cylinder. Mark the height of the flattened Play-Doh on the cylinder. Determine the relationship between the height of the cylinder and the height of the Play-Doh. Using this data, what inferences can you make about the relationship between the volume of the sphere and the volume of the cylinder? What might the formula be for finding the volume of a sphere?

The textbook formula for the volume of a sphere is: . Is this different from the formula you predicted? If so, why might this be? Now remove the Play-Doh from the cylinder. Take a sheet of paper and roll it into a cone so the point of the cone touches the bottom of the cylinder. Tape the paper so it holds this shape, and then trim the paper so it is the same height as the cylinder.

Fill the cone to the top with rice, and then empty it into the cylinder. Repeat this as many times as necessary until the cylinder is full of rice. Compare the volume of the cone with the volume of the cylinder to determine a relationship between the two. Use this information to make an inference about the volume formula for a cone.

If a cone, cylinder, and sphere have the same radius and same height, what is the relationship between the volumes of these three shapes?