# Area

### Topics

Quick! What's the most famous square in the world? Times Square? Tiananmen Square? Madison Square Garden? Square root? Spongebob Squarepants?

We hate to burst your bubble, but none of those is actually a square.

**Squares** are quadrilaterals that have four right angles and four sides of equal length. They're also a special type of rectangle. (Remember, all squares are rectangles, but not all rectangles are squares.)

But wait. If they're rectangles, can't we just use the area formula for rectangles on squares? Indeed we can, but we run into a slight problem. Which side is length and which side is width?

It's sort of a trick question because it doesn't matter. Since all sides of a square are the same, the length and width are exactly the same. Since it's pointless to have two identical sides with different names, we call both the length and width of a square the **side**. After all, separate but equal is inherently unequal and we're all for equality.

So when we apply the formula *A* = *lw* to squares, we can say that *l* = *w* = *s*, and end up with *A* = *s*^{2}. Simple enough, right?

### Sample Problem

Find the area of the square.

Here we have a square with side length *s* = 4 units. (If we weren't sure this was a square, we could also say that *l* = *w* = 4 units for the rectangle. Since they're equal though, it's obviously a square.) In any case, we can use the area formula for a square.

*A* = *s*^{2}*A* = (4 units)^{2}*A* = 16 units^{2}

As far as the area of rectangles goes, that's really all there is to say. You'll probably encounter problems that require you to find *l*, *w*, or *s* before you can calculate *A*. Or maybe they'll ask you to solve for one of the sides given the area. There are many ways to spin these problems, but as long as you can identify the formula you need and the variable you're looking for, there shouldn't be a problem.

Except for the sample problems, of course.

### Sample Problem

Find the area of a square with the same perimeter as the rectangle below.

The perimeter is the distance around a 2D shape. In the case of a rectangle, it's just the sum of all the side lengths.

*P* = 8 m + 10 m + 8 m + 10 m = 36 m

So the mystery square whose area we're looking for also has a perimeter of 36 m.

Since a square has sides of equal lengths, we know that its perimeter will just be all its sides added together, or 4 times the length of any side. We know the perimeter has to be the same, or 36 m, so we can substitute that to find *s*.

*P* = 4*s*

36 m = 4*s*

s = 9 m

Now that we know the side length of the square, we can solve for the area.

*A* = *s*^{2}*A* = (9 m)^{2} = 81 m^{2}

The area of the square is 81 m^{2}. Notice that the square has the same perimeter as the initial rectangle, but the area is larger.