Inscribed Solids

　　An inscribed solid is a solid placed inside another solid, with the edges of the two solids touching. The figures below are, from left to right, a cylinder inscribed in a sphere, a sphere inscribed in a cube, and a rectangular solid inscribed in a sphere.

　　Math IC questions that involve inscribed solids don’t require any techniques other than those you’ve already learned. These questions do require an ability to visualize inscribed solids and an awareness of how certain line segments relate to both of the solids in a given figure.　　Most often, an inscribed-solid question will present a figure of an inscribed solid and give you information about one of the solids. For example, you may be given the radius of a cylinder, and then be asked to find the volume of the other solid, say a rectangular solid. Using the figure as your guide, you need to use the radius of the cylinder to find the dimensions of the other solid so that you can answer the question. Here’s an example:

　　In the figure below, a cube is inscribed in a cylinder. If the length of the diagonal of the cube is 4 and the height of the cylinder is 5, what is the volume of the cylinder?

　　The formula for the volume of a cylinder is πr2(h). The question states that h = 5, but there is no value given for r. So in order to solve for the volume of the cylinder, we need to first find the value of r.　　The key step in this problem is to recognize that the diagonal of a face of the cube is also the diameter, or twice the radius, of the cylinder. To see this, draw a diagonal, d, in either the top or bottom face of the cube.

　　In order to find this diagonal, which is the hypotenuse in a 45-45-90 triangle, we need the length of an edge of the cube, or s. We can find s from the diagonal of the cube (not to be confused with the diagonal of a face of the cube), since the formula for the diagonal of a cube is s where s is the length of an edge of the cube. The question states that the diagonal of the cube is 4 so it follows that s = 4. This means that the diagonal along a single face of the cube is 4 (using the special properties of a 45-45-90 triangle). Therefore, the radius of the cylinder is　Plug that into the formula for the volume of the cylinder, and you get π　(2)2　5 = 40π.　　Helpful Tips　　Math IC questions involving inscribed solids are much easier to solve when you know how the lines of different solids relate to one another. For instance, the previous example showed that when a cube is inscribed in a cylinder, the diagonal of a face of the cube is equal to the diameter of the cylinder. The better you know the rules of inscribed solids, the better you’ll do on these questions. So without further ado, here are the rules of inscribed solids that most commonly appear on the Math IC.　　Cylinder Inscribed in a Sphere　　The diameter of the sphere is equal to the diagonal of the cylinder’s height and diameter.

　　Sphere Inscribed in a Cube　　The diameter of the sphere is equal to the length of the cube’s edge.

　　Sphere Inscribed in a Cylinder　　Both the cylinder and the sphere have the same diameter and radius.