## SAT SUBJECT TEST MATH LEVEL 1

## SOLID AND COORDINATE GEOMETRY

## CHAPTER 12 Solid Geometry

### CYLINDERS

A ** cylinder** is similar to a rectangular solid except that the base is a circle instead of a rectangle. To find the volume of a rectangular solid, we multiply the area of its rectangular base,

*w*, by its height,

*h*. For a cylinder, we do exactly the same thing. The volume of a cylinder is the area of its circular base,

*r*

^{2}, times its height,

*h*. The surface area of a cylinder depends on whether you are envisioning a tube, such as a straw without a top and bottom, or a can, which has both a top and bottom.

**Key Fact K4**

• **The formula for the volume, V, of a cylinder whose circular base has radius r**

**and whose height is**

*h***is**=

*V*

*r*^{2}*h*.• **The formula for the surface area, A, of the side of a cylinder is the product of the circumference of its circular base and its height: A** =

**2**π

*rh*.• **The areas of the top and bottom of a cylinder are each** π** r^{2}, so the total surface area of a cylindrical can is 2**π

**+**

*rh***2**π

*r*^{2}.**EXAMPLE 3:** You can roll an 8 x 12 rectangular piece of paper into a cylinder in two ways. You could tape the 8-inch sides together, or you could tape the 12-inch sides together. Note that these cylinders do *not* have the same volume.

Cylinder I

Cylinder II

In cylinder I, *C* = 12 2π*r* = 12 *r* = , and so

In cylinder II, *C* =8 2π *r* = 8 *r* = , and so