# Geometry Tutorial for SAT and GRE Test Preparation

Geometry is a little different from arithmetic and algebra. Geometry is all about spatial reasoning – understanding points, lines and figures, and how they relate to each other. You’ll need to master some new concepts, including perimeter, area, volume, and coordinate graphing. Many of these concepts are expressed in formulas, and working with formulas requires good arithmetic and algebra skills. So while geometry does involve new concepts, it also demands the same basic techniques as other areas of mathematics.

Basic Concepts

Let’s start by defining a few important concepts like perimeter, area and volume. Then we will apply these concepts to squares, rectangles, circles and triangles.

After that, we’ll talk about lines and some of their properties, in particular the relations between the angles formed by combinations of intersecting and parallel lines.

Finally we’ll look at some coordinate geometry: coordinates, slopes and distances.

That’s a lot, so let’s get started…

**Perimeter**

Take a look at your monitor. If you take your finger and trace a path around the outer edge, how far will your finger travel?

To measure that distance, you take a ruler and add up all the straight edge distances. The total length is the perimeter of the monitor.

Because it is the total length of a series of straight lines, perimeter is something you report in units of length, be they inches, meters, miles or whatever.

**Area**

Look at your monitor again. This time, let’s ask how much cloth you would need to cover the side facing you.

To answer this question you would take your ruler and measure the vertical height and the horizontal width.

For example, my monitor is 10 inches high and 13 inches wide. So the area of the face of my monitor is:

**Area = height × width** = 10 inches × 13 inches = 130 inches²

Because area is a product of two dimensions, it is expressed in square units of length.

**Volume**

Ok, one last look at your monitor, this time requiring a little more imagination. Suppose your monitor were hollow. How big a space is there inside? How much water would it take to fill that space? (please don’t try it!)

Well, you’ve probably caught on to the pattern. You would take your ruler and measure the height and width and depth.

Let me measure mine…

Ok, mine is 10 inches high, 13 inches wide and 2 inches deep (it’s a flat screen monitor).

The volume is:

**Volume = height × width × depth**

= 10 inches × 13 inches × 2 inches = 260 inches³

Volume, being the product of three dimensions, is measured in cubic units of distance.

Notice that you can think of volume as the area of one surface multiplied by another dimension. For example, the volume of the monitor is the area facing you times the depth.

Volume = [Area of side facing you] × Depth

= 130 inches² × 2 inches

= 260 inches³

Now that we have the basic concepts of perimeter, area, and volume down, let’s start looking at how we can apply these concepts to different kinds of shapes.