Explainer: Areas of Rectangles and Squares

In this explainer, we will learn how to find the areas of rectangles and squares using a formula with fractions and decimals and solve real-life problems.

Definition: Area, Length, Width, Square Unit

  • Area is the number of square units needed to cover a surface.
  • The length of a rectangle is the length of its longest side, and the width is the length of its shortest side.
  • The unit square is a square whose sides measure 1 length unit. The area of the unit square is equal to 1 square unit.

We measure area in square units; depending on the unit used to measure the length, these could be square centimeters, square inches, square miles, or something else. We already know that the area of a rectangle can be found by counting how many unit squares compose the shape.

For example, if we have a rectangle whose length is 3 units and width is 2 units, then there are 2 groups of 3 square units, or 3 groups of 2 square units.

Hence, we multiply the length by the width to find the area, 𝐴=2×3=6.squareunits

The Area of a Rectangle

The area 𝐴 of a rectangle is equal to its length 𝑙 multiplied by its width 𝑤.

The formula for the area is 𝐴=𝑙𝑤.

Sometimes, instead of length and width, the dimensions are called base and height. So, if a rectangle has height and base length 𝑏, then its area is 𝐴=𝑏.

Remember that area is measured in square units.

For example, the area of this rectangle is 4×3=12 square units.

Example 1: Finding the Area of a Rectangle

Find the area of rectangle 𝐴𝐵𝐶𝐷.

Answer

Use the formula for the area of rectangle: areaofrectanglelengthwidthcmcmcm=×=8×6=48.

Once we know the area of a rectangle, we can easily deduce the formula for the area of a square. Since a square is also a rectangle, its area is equal to 𝐴=𝑙𝑤 where the length is equal to the width; so for a square, 𝐴=𝑙×𝑙.

Hence, we can calculate the area of a square using just the length of one of the sides.

The Area of a Square

The area 𝐴 of a square is equal to the square of the length 𝑙 of one of its sides.

The formula for the area is 𝐴=𝑙.

Remember that area is measured in square units.

For example, the area of this square is 4×4=4=16 square units.

Example 2: Finding the Area of a Square

Find the area of the square.

Answer

Use the formula for the area of a square: areaofsquaresidelengthsidelengthcmcmcm=×=5×5=25.

We will finish by looking at two more examples.

Example 3: Solving Word Problems Using the Formula for the Area of Rectangles

The size of youth soccer fields depends on the age group of the players. Use the table to determine the difference between the area of a soccer field for under 14s compared to the field for under 12s.

Age GroupUnder 10sUnder 12sUnder 14s
Field Length (yards)7080100
Field Width (yards)405060

Answer

We can start with a sketch of what we know. From the table, we have that the under 12s play on a rectangular field which measures 80 yards by 50 yards, and the under 14s play on a rectangular field which measures 100 yards by 60 yards.

First, we will calculate the area of each field using that the area of a rectangle is equal to the length multiplied by the width. Note also that since the lengths are given in yards (yd), the unit for the area will be square yardsyd.

The area of the under 12s field is 80×50=4,000.yd

The area of the under 14s field is 100×60=6,000.yd

Then, we calculate the difference in the areas as 6,0004,000=2,000.yd

Example 4: Solving Word Problems Using the Formula for the Areas of Rectangles and Squares

A rectangular carpet measuring 4 m by 3 m is placed in a square room of length 9 m. What area of the floor is not covered by the carpet?

Answer

Step 1: Find the area of the rectangular carpet: areaofrectanglelengthwidthm=×=4×3=12.

Step 2: Find the area of the square floor: areaofsquaresidelengthitselfm=×=9×9=81.

Step 3: Find the area of the floor that is uncovered.

Subtract the area of the rectangular carpet from the area of the square floor: uncoveredaream=8112=69.

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