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Study Guide

We've already discussed the different types of quadrilaterals. Hopefully we haven't forgotten what we learned, because they've come back to haunt us. Big time.

Since rectangles (and squares, which as you know are also rectangles) are among the simpler types of quadrilaterals, we'll start there. The trouble is that rectangles are pretty cocky. It's tough to deal with shapes that think they're always right. Just because they *are* doesn't mean they have to rub it in everyone's face.

As it turns out, though, these right angles come in handy. We can find the area of a **rectangle** by simply multiplying the lengths of both the sides together. The longer side is called the **length** for obvious reasons, and the shorter side is the **width**.

The area of a rectangle, then, is given by the formula *A* = *lw*, where *l* is the length and *w* is the width. Again, for obvious reasons.

Van Gogh is super tired of painting, so he offers you $100 to paint the wall of his house. He provides you with a blueprint, and you discover that the wall is rectangular and is 30 feet by 15 feet. How much total wall do you have to paint?

To find the area of the rectangular wall, we just have to multiply its dimensions together. Length times width.

*A* = *lw*

We know that the length is 30 feet and the width is 15 feet. So we can replace *l* with 30 ft and *w* with 15 ft.

*A* = 30 ft × 15 ft*A* = 450 ft^{2}

That's a decent-sized wall, and $100 won't cut it. That Van Gogh; such a cheapskate.

What is the area of this rectangle?

All we need for the area of a rectangle is the length and the width. Here, we have the width, but we're missing the length. Bummer. But we *do *know the diagonal of the rectangle, and it forms a right triangle with one width and length of the rectangle.

We've talked about triangles (and especially right triangles) enough to spot the Pythagorean Theorem from a mile away. In this particular case, we know that *a* = 5, *b* = *l*, and *c* = 13.

*a*^{2} + *b*^{2} = *c*^{2}

5^{2} + *l*^{ 2} = 13^{2}

25 + *l*^{ 2} = 169*l*^{ 2} = 144*l* = 12

Now that we know the length and the width, we can solve for the area of the rectangle.

*A* = *lw**A* = 12 × 5 = 60 units^{2}

### Area of Squares

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 squares 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 mSo 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*=*s*+*s*+*s*+*s**P*= 4*s*

36 m = 4*s*

s = 9 mNow 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.

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