# Area

### Topics

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 triangle, then, is given by the formula *A* = *lw*.

### Sample Problem

Upon request, Van Gogh finally provides you with the blueprint to his house and you discover that the wall 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.

### Sample Problem

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. The diagonal of the rectangle is given, 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, though, 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}