Area of Rectangles and Parallelograms - Sixth Grade Mathematics

Area is the quantitative measure of two-dimensional surface enclosed within a boundary, measured in square units.

The Foundation of Square Units and Rectangles

Area measures how many standard unit squares (such as square centimeters, square inches, or square meters) completely tile a flat region without gaps or overlaps.

Tiling a Rectangle (Length 5 units, Width 3 units):
         +---+---+---+---+---+
         | 1 | 2 | 3 | 4 | 5 |  Row 1 (5 unit squares)
         +---+---+---+---+---+
         | 6 | 7 | 8 | 9 |10 |  Row 2 (5 unit squares)
         +---+---+---+---+---+
         |11 |12 |13 |14 |15 |  Row 3 (5 unit squares)
         +---+---+---+---+---+
         Area = 5 columns * 3 rows = 15 square units.

Formulas for Rectangles and Squares:
Rectangle:   Area = length * width        (A = l * w)   or   Area = base * height (A = b * h)
Square:      Area = side * side = side^2  (A = s^2)

Square units are designated with an exponent of 2: square meters is written m^2, square centimeters is cm^2, and square feet is ft^2.

Deconstructing Parallelograms to Rectangles

A parallelogram is a four-sided polygon with opposite sides parallel and equal. Its area formula is derived directly from a rectangle through a geometric process of cutting and rearranging.

Transforming a Parallelogram into a Rectangle:
                Base b
         +-----------------+
        /|                 /
       / |                /
    h /  |               / h
     /   |              /
    +----+-------------+
      cut      Base b

Take the right triangle on the left (cut along perpendicular height h):
Shift it across to the right side!

         +-----------------+-----+
         |                 |    /|
         |                 |   / |
       h |                 |  /  | h
         |                 | /   |
         +-----------------+-----+
                Base b

The resulting shape is an exact RECTANGLE with base b and height h!
Area of Parallelogram = base * height   (A = b * h)

Perpendicular Height vs. Slant Height

A critical mistake in calculating parallelogram area is using the slanted side length instead of the perpendicular height.

Identifying the Correct Dimensions:
                   Base b = 10 cm
             +-----------------------+
            / :                     /
  Slanted  /  :                    /
   Side   /   : Perpendicular     /  Slanted Side = 6 cm
 s = 6 cm/    : Height h = 5 cm  /
        /     :                 /
       +------+----------------+
              Base b = 10 cm

Area = b * h = 10 cm * 5 cm = 50 cm^2.
(The slant length of 6 cm is ONLY used for perimeter, NEVER for area!)

The height h must always form a 90-degree right angle with the chosen base b.

Comparing Area and Perimeter

Two shapes can have the exact same perimeter while possessing completely different areas. For example, a 1 by 9 rectangle has perimeter 2(1 + 9) = 20, but area 1 * 9 = 9 square units. A 5 by 5 square has the same perimeter 2(5 + 5) = 20, but its area is 5 * 5 = 25 square units. For any fixed perimeter, the shape closest to a square encloses the greatest area.

Exercises

Exercise 1. Calculate the area and perimeter of each figure: (a) A rectangle with length 12.5 meters and width 4 meters; (b) A square with side length 9 cm; (c) A parallelogram with base 16 inches, perpendicular height 7 inches, and slant side 8.5 inches.

Exercise 2. A rectangular living room carpet covers an area of 216 square feet. If the width of the room is 12 feet, what is its length?

Exercise 3. A parallelogram has an area of 84 square centimeters. If its perpendicular height is 7 cm, what is the length of its base?

Exercise 4. A city park is designed in the shape of a parallelogram with a base of 150 meters and a perpendicular height of 80 meters. (a) What is the total area of the park? (b) If grass sod costs $2.50 per square meter, how much will it cost to cover the entire park?

Solutions

Solution to Exercise 1. For (a), Area = l * w = 12.5 * 4 = 50 m^2; Perimeter = 2(12.5 + 4) = 2(16.5) = 33 meters. For (b), Area = s^2 = 9^2 = 81 cm^2; Perimeter = 4(9) = 36 cm. For (c), Area = b * h = 16 * 7 = 112 in.^2; Perimeter = 2(16 + 8.5) = 2(24.5) = 49 inches.

Solution to Exercise 2. Using A = l * w: 216 = l * 12. Divide both sides by 12: l = 216 / 12 = 18 feet. The length of the living room is 18 feet.

Solution to Exercise 3. Using A = b * h: 84 = b * 7. Divide both sides by 7: b = 84 / 7 = 12 cm. The base length is 12 centimeters.

Solution to Exercise 4. For (a), Area = b * h = 150 * 80 = 12,000 m^2. For (b), Total Cost = 12,000 m^2 * $2.50/m^2 = $30,000.