Connecting Area to Arrays: Length x Width - Third Grade Mathematics

Counting individual unit squares one by one works well for small figures, but counting hundreds of tiny tiles on a large floor would take all day. Because unit squares inside a rectangle are arranged in perfect rows and columns, calculating area connects directly to the multiplication arrays you mastered earlier! In this chapter, you will discover the foundational formula for the area of a rectangle—Length times Width—and learn to calculate area with speed and precision.

The Connection Between Arrays and Rectangles

Recall what an array looks like: An array of 4 rows of 6 counters has 4 x 6 = 24 counters in all.

Now, replace each counter with a 1-unit square:

       <------------- 6 units wide ------------->
      +-----+-----+-----+-----+-----+-----+
      | [ ] | [ ] | [ ] | [ ] | [ ] | [ ] |  Row 1 (6 squares)
      +-----+-----+-----+-----+-----+-----+
      | [ ] | [ ] | [ ] | [ ] | [ ] | [ ] |  Row 2 (6 squares)
      +-----+-----+-----+-----+-----+-----+
      | [ ] | [ ] | [ ] | [ ] | [ ] | [ ] |  Row 3 (6 squares)
      +-----+-----+-----+-----+-----+-----+
      | [ ] | [ ] | [ ] | [ ] | [ ] | [ ] |  Row 4 (6 squares)
      +-----+-----+-----+-----+-----+-----+

Notice what just happened:
- The number of rows is 4 (the height or Width).

- The number of squares in each row is 6 (the base or Length).
Instead of counting 1, 2, 3... 24, you can simply multiply: 4 rows x 6 squares per row = 24 square units!

The Area Formula for Rectangles

Mathematicians express this array relationship using a universal formula:

Formula: Area = Length x Width
Abbreviation: A = L x W

                   Length (L)
         +----------------------------+
         |                            |
Width (W)|       Area = L x W         |
         |                            |
         +----------------------------+

The Area Formula for Squares

In a square, the length and width are equal (both are called Side, s).

Formula: Area of a Square = Side x Side
Abbreviation: A = s x s

If a square has a side length of 5 cm: Area = 5 cm x 5 cm = 25 square centimeters (25 sq cm).

Always Label Units Correctly!

In mathematics, units matter tremendously:
- Length and perimeter are measured in linear units: inches, feet, cm, meters.

- Area is ALWAYS measured in square units: square inches, square feet, square cm, square meters.
Writing "24 cm" when you mean "24 square cm" is like confusing a piece of string with a sheet of paper!

Worked Examples

Let us solve several area problems using the formula.

Example 1: Bedroom Rug Area

A bedroom rug is 8 feet long and 5 feet wide. What is the area of the rug? A = L x W A = 8 ft x 5 ft = 40 square feet.

Example 2: Poster Board Area

A student poster board is 9 inches long and 7 inches wide. What is its area? A = 9 in x 7 in = 63 square inches.

Example 3: Finding an Unknown Side from Area

A rectangle has an area of 36 square meters. Its width is 4 meters. What is its length? Since Area = Length x Width: 36 = Length x 4 Think: What number times 4 equals 36? (36 ÷ 4 = 9). Length = 9 meters!

Chapter Practice Exercises

  1. Calculate the area of each rectangle: a. Length = 7 cm, Width = 4 cm b. Length = 8 inches, Width = 6 inches c. Length = 10 meters, Width = 3 meters d. Length = 9 feet, Width = 5 feet
  2. Calculate the area of each square: a. Side length = 6 inches b. Side length = 8 meters c. Side length = 4 cm
  3. A rectangular garden has an area of 48 square feet. If the length is 8 feet, what is the width of the garden?
  4. A square playground has an area of 49 square yards. What is the length of one side of the playground?
  5. A tabletop measures 4 feet wide and 6 feet long. a. What is the perimeter of the tabletop? b. What is the area of the tabletop? c. Notice the different units for perimeter and area!

Solutions and Step-by-Step Answers

  1. Rectangle areas: a. 7 cm x 4 cm = 28 square cm. b. 8 in x 6 in = 48 square inches. c. 10 m x 3 m = 30 square meters. d. 9 ft x 5 ft = 45 square feet.
  2. Square areas: a. 6 in x 6 in = 36 square inches. b. 8 m x 8 m = 64 square meters. c. 4 cm x 4 cm = 16 square cm.
  3. Garden width: Area = 48 sq ft. Length = 8 ft. Width = 48 ÷ 8 = 6 feet.
  4. Playground side: Area of square = 49 sq yd. Side x Side = 49 -> 7 x 7 = 49. The side length is 7 yards.
  5. Tabletop: a. Perimeter = 4 + 6 + 4 + 6 = 20 feet. b. Area = 4 x 6 = 24 square feet. c. Perimeter is 20 feet (linear distance around), while area is 24 square feet (flat surface covered).