# Area

### Topics

## Introduction to :

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}

#### Example 1

What is the area of a rectangle with length of 7 miles and a width of 3 miles? |

#### Example 2

If the area of a rectangle is 140 cm |

#### Example 3

A rectangle has an area of 800 in |

#### Example 4

A rectangle has side lengths of 8 yards and 18 yards. What are the side lengths of a square with the same area? |

#### Exercise 1

What is the area of a rectangular baking pan that is 10 inches by 12 inches?

#### Exercise 2

Solve for the area of the rectangle below.

#### Exercise 3

What area of your bedroom floor does your twin size bed (39 inches wide and 75 inches long) take up?

#### Exercise 4

You're (finally) off to college! Your dorm room has an extra-long twin size bed, which is 5 inches longer than your twin bed at home. How much more area do you get with an extra-long twin as compared to a twin size bed (39 by 75 inches)?

#### Exercise 5

You are trying to set the table for Thanksgiving, and you have a square napkin with side length of 8 inches. What is the total area of the napkin?

#### Exercise 6

The square napkin with side length 8 inches doesn't fit at the place settings unless it is folded into fourths. After folding the napkin in half one way and folding it in half again the other way, what is the area?

#### Exercise 7

The town square in Tiny-ville is 169 Tiny-meters^{2}. What is the length of a side?

#### Exercise 8

A square TV screen has a diagonal of 3 feet. What is the area of the screen?

#### Exercise 9

You decide to get your boyfriend a box of fudge for his birthday. At the Fudge-tastic store, you can buy a 4 × 3 inch piece of fudge for $6. At the Fudge-licious store, you can buy a 1 foot × 1 inch bar of fudge for $6. If the fudge pieces are of equal thickness, which store should you buy from for the better deal?

#### Exercise 10

Your new mansion has no pool, and summer is fast approaching. If you have backyard space that is 250 yards by 250 yards, and you want to build a pool that's 100 by 50 yards, what percentage of your backyard space will the pool occupy?