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# Algebraic Expressions

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# Four-Sided Shapes

Don't you love it when you can rely on common sense rather than having to memorize complicated formulas? Yeah, us too. Well then, you will absolutely adore four-sided shapes. You might even marry them. You and four-sided shapes, sitting in a tree, K-I-S-S-I-N-G.

To determine the distance all the way around a four-sided shape, you simply add the lengths of the four sides. If you remember this step, you can figure out whatever formula you need by drawing a picture of the shape and thinking for 30 seconds. If that makes your brain throb, you can think for 15 seconds, take a quick snack break, then come back and think for 15 more.

A **parallelogram** is a four-sided figure in which opposite sides are parallel. Opposite sides will be equal, and opposite angles will be equal as well. Equality reigns supreme here in Paralleloland. My, that is a lot of "l"s.

The perimeter is the sum of all four sides. We could write this as the formula *P* = 2*b* + 2*c *where *P* stands for perimeter. However, it is more important (and simpler) to remember that the perimeter is the sum of all four sides than to commit this formula to memory. You have got enough numbers and symbols floating around in that noggin of yours. Let's free some space for a change.

The area of a parallelogram is its height times its base. To see why that is, chop off the left corner of the parallelogram (watch your fingers):

Then move it over to the right side:

Now we have a rectangle that has the same area as the parallelogram. The area of this rectangle is *bh*, and therefore, the area of the parallelogram is also *bh*. What a copycat. If we let *A *be the area of the parallelogram, we can express this fact via the formula

*A *= *bh*

A **rectangle** is merely a particular type of parallelogram: it has right angles at all four corners. Since a rectangle is a parallelogram, opposite sides will still have the same length. Note that this fact does not work in reverse—not all parallelograms are necessarily rectangles. Some of them are Virgos.

The perimeter *P* of a rectangle is given by the formula

*P* = 2*l* + 2*w,*

and the area *A* of a rectangle is given by the formula

*A* = *lw.*

A **square** is a rectangle with all four sides the same length. Squares live in very strict, cookie-cutter communities. Also, they are restricted in the number of guests they can have in their backyard on weeknights.

The perimeter *P* of a square is

*P* = 4*s,*

and the area *A* of a square is

*A* = *s*^{2}.

A **trapezoid** is a four-sided figure with *one* pair of opposite sides parallel. The sides that are parallel to each other are called the **bases** of the trapezoid. That may be a little confusing, as we are used to thinking of a base as being only on the bottom of something, but this feller does indeed have a base on top as well. He's got all his bases covered.

In the picture below, *b*_{1} and *b*_{2} are the bases.

As with a square, the perimeter is the sum of all four sides. While we could write a formula for perimeter in terms of the names of the sides, it wouldn't make any more sense than writing "the perimeter is the sum of all four sides," and let's not go there. Formulas are cool and all, but straightforward concepts expressed in plain English are the bomb.

The area *A* of the trapezoid is given by

.

To see the justification for this formula, read about the distributive property and simplifying expressions. However, since that will involve a lot of clicking and reading, our hunch is that you will assume we know what we are talking about. That's perfectly fine, too. Good thing we don't get off on pulling your leg.