**Middle School Number Sense Lesson 42: Rectangles**

Most mathletes have been working with rectangles since we were little. We learned the formulas for the area and perimeter of a rectangle soon after we mastered squares. For number sense purposes, area and perimeter are the main concepts dealing with rectangles--which is pretty basic. But we will see a little more complexity in rectangle questions--especially on high school tests. This concept appeared

**11 times**in middle school this year, with a median placement at

**question # 36**.

**Number Dojo Level: 147**

Let's review the formulas for area and perimeter.

**L**stands for

**length**and

**W**stands for

**width**.

**Area: L x W****Perimeter: 2(L + W)**

**Example 1: The area of a rectangle with length 22 and width 18 is ___**

- Area = L x W = 22 x 18.
- Using
**DiffSquares1**, 22 x 18 =**396**.

**Example 2: If the area of a rectangle is 42 and the length is 10, then the width is ___**

- Area = L x W. 42 = 10 x W.
- W = 42 ÷ 10 =
**4.2**.

**Example 3: Find the length of a rectangle with perimeter 48 and width 8.**

- Perimeter = 2(L + W). 48 = 2(L + 8).
- 48 = 2L + 16, so 2L = 48 - 16 = 32.
- 2L = 32, so L =
**16**.

**Example 4: The area of a rectangle is 160. If its length is 16, then its perimeter is ___**

- Area = L x W. 160 = 16 x W. W = 160 ÷ 16 = 10.
- Perimeter = 2(L + W) = 2(16 + 10) = 2(26) =
**52**.

**Example 5: If a rectangle of area 40 has width 5, then the perimeter of the rectangle is ___**

- Area = L x W. 40 = L x 5. L = 40 ÷ 5 = 8.
- Perimeter = 2(L + W) = 2(8 + 5) = 2(13) =
**26**.

**Example 6: Find the ratio of the perimeter of a 3" x 6" rectangle to its area. ___**

- Area = L x W = 6 x 3 = 18.
- Perimeter = 2(L + W) = 2(6 + 3) = 2(9) = 18.
- Ratio = Perimeter ÷ Area = 18 ÷ 18 =
**1**.

*Another concept related to rectangles deals with right triangles*. (If you draw one diagonal of a rectangle, the diagonal divides the rectangle into two identical right triangles). The

**Pythagorean Theorem**states that for right triangles:

**A**

^{2}+ B^{2}= C^{2}Or, if you think in terms of the parts of the rectangle, with

**D**representing the

**diagonal**,

**L**.

^{2}+ W^{2}= D^{2}**Example 7: Find the diagonal of a rectangle with length 15 and width 8**.

1. D

^{2}= L

^{2}+ W

^{2}= 15

^{2}+ 8

^{2}.

2. D

^{2}= 225 + 64 = 289.

3. D =

**17**.

**Here's a free worksheet to help you practice Rectangle:**

rectangle.pdf |

**Up Next for Middle School: Trapezoid**