Three-Dimensional Geometry

Three-dimensional geometry, or solid geometry, adds some depth to plane geometrical figures.

1. Rectangular Solids

You make a rectangular solid by taking a simple rectangle and adding depth. A rectangular solid is also known as a right rectangular prism because it has 90-degree angles all around. Prisms have two congruent polygons on parallel planes that are connected to each other by their corresponding points. The two connected polygons make up the bases of the prism.

A rectangular solid has three dimensions: length, height, and width.

The volume (V) of a rectangular solid is a measure of how much space it occupies. You measure the volume of an object in cubic units. The formula for the volume of a rectangular solid is simply its length (l) times its width (w) times its height (h). Another way of saying this formula is that the volume is equal to the base times the height.

V = lwh

You can find the surface area (SA) of a rectangular solid by simply figuring out the areas of all six sides of the object and adding them together. First you find the area of the length (l) times height (h), then the area of length times width (w), and finally width times height. Now multiply each of these three area measurements times 2 (after you find the area of one side, you know that the opposite side has the same measurement). The formula for the surface area of a rectangular solid is

SA = 2(lh + lw + wh)

2. Cubes

You can use the same formulas you used with rectangular solids to find the area and volume of a three-dimensional square, called a cube. Because all the faces on a cube are perfect squares, you can find its measurements with some simple formulas.

The volume of a cube with an edge a is:

V = a3

The surface area of a cube is the area of one side times 6:

SA = 6a2

The diagonal of a face on a cube (a square) measures √2a.

The diagonal of a cube measures √3a.

3. Cylinder

A cylinder is a circle that grows straight up into the third dimension to become the shape of a can of soda. The bases of a cylinder are two congruent circles on different planes. The cylinders you see on the GMAT are right circular cylinders, which means that the line segments that connect the two bases are perpendicular to the bases.

All the corresponding points on the circles are joined together by line segments. The line segment connecting the center of one circle to the center of the opposite circle is called the axis.

A right circular cylinder has the same measurements as a circle. That is, a right circular cylinder has a radius, diameter, and circumference. In addition, a cylinder has a third dimension: its height, or altitude.

To get the volume of a right circular cylinder, first take the area of the base (a circle), and multiply by the height (h) of the cylinder.

V = πr2h

If you want to find the total surface area of a right circular cylinder, you have to add the areas of all the surfaces. Imagine taking a soda can, cutting off the top and bottom sections, and then slicing it down one side. You then spread out the various parts of the can. If you measure each one of these sections, you get the total surface area.

TSA = 2πrh + 2πr2