Area of Rectangles

A rectangle is a special case of a quadrilateral, which is a four-sided plane figure. All 4 internal angles of a rectangle are right angles. A book, a football field, a window, a traveling suitcase are all examples of rectangles.

Now suppose you want to calculate the total space covered by a football field. Then, you would need to know how to calculate the area of a rectangle.

A quadrilateral with 2 pairs of parallel opposite sides is called a parallelogram. Since all angles of a rectangle are right angles, it turns out that the opposite pairs of sides of a rectangle are always parallel. This makes every rectangle a parallelogram. In fact, a rectangle is considered a special type of parallelogram.

Area of rectangles: Formula

The area of a rectangle is the measure of the amount of space enclosed by the rectangle. It is calculated by multiplying the length of the rectangle by its width. The area of a rectangle is calculated by multiplying its length by its width. Consider the following rectangle.

Finding the area of the Rectangle through this illustration example

The area of a rectangle is given by the formula:

\[Area = b \times h\]

where b = length of base, h = length of height

Now the value, b, is the length of the side AB, which is considered to be the base here. Conventionally, one of the longer sides of the rectangle is taken to be the base, and one of the sides perpendicular to the base is considered to be the height. In this rectangle, the height is equal to the length of AD.

Special case: Formula for the area of a square

A square is a special case of a rectangle. In addition to all 4 internal angles being right angles, all 4 sides of a square are equal.

An example of the area of the Square.

Look at the above square and recall the formula for the area of a rectangle: \[Area = base \times height.\]

Since all 4 sides of a square are equal, the base and height are equal. Just knowing the length of the sides of a square is enough to calculate its area. Thus, in the case of a square the formula can be reduced to:

\[Area = length\,of\,side \times length\,of\,side = (length\,of\,side)^2\]

Area of rectangles: Square units

When considering the area of a figure, remember that area is measured in square units, such as square centimeters (cm²), square feet (ft²), square inches (in²), etc.

If you are unfamiliar with the square unit, it is helpful to consider the concept as it is represented visually in the figure below. Consider how many square units are needed to exactly and exhaustively cover the entire surface of a closed figure. This amount is the figure's area.

Square units, Jurgensen & Brown–Geometry

Area of rectangles: Example problems

The following examples show how to find the area of a rectangle.

Example 1: Suppose you have a rectangle with a length of 10 units and a width of 5 units. To find the area:

\[Area=10\,units\times5\, units=50\, square\, units\]

Example 2: Imagine a garden plot shaped as a rectangle with a length of 15 meters and a width of 8 meters. To determine the area:

\[Area=15\,meters\times8\, meters=120\, square\, meters\]

Example 3: Consider a rectangular swimming pool with a length of 25 meters and a width of 10 meters. To find the area:

\[Area=25\,meters\times10\, meters=250\, square\, meters\]

Here is a more in-depth breakdown on how to find the area of a rectangle:

If you are given the length of 1 of the sides (base or height) of a rectangle and the length of the diagonal, you can calculate the unknown side length (height or base) using the Pythagoras' Theorem. The Pythagoras' theorem states that in a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other 2 sides.

The following figure shows how the diagonal of a rectangle divides it into 2 right angled triangles, thus allowing us to use the Pythagoras' theorem. Then, once both the base and height of the rectangle are known, the area can be calculated.

The diagonal of a rectangle divides it into 2 right angles triangles.

Area of rectangles with fractions

If the length and width of a rectangle are given as fractions, you can still calculate its area by multiplying these fractions.

Let's break it down with an example:

Suppose the length \( b \) of a rectangle is \( \frac{3}{4} \) units and the width \( h \) is \( \frac{2}{5} \) units.

To find the area \( A \) of the rectangle: \[ A = b \times h \] Substitute in the given fractions: \[ A = \frac{3}{4} \times \frac{2}{5} \] To multiply the fractions: \begin{align*} \text{Multiply the numerators:} & \quad 3 \times 2 = 6 \\ \text{Multiply the denominators:} & \quad 4 \times 5 = 20 \\ \end{align*} This gives: \[ A = \frac{6}{20} \] Simplifying the fraction by dividing both the numerator and the denominator by their greatest common divisor: \[ A = \frac{3}{10} \] Thus, the area of the rectangle is \( \frac{3}{10} \) square units.

Perimeter and area of rectangles

The perimeter and area are two fundamental properties of a rectangle.

Perimeter of a Rectangle: The perimeter is the total distance around the rectangle, or the sum of all its sides. Since opposite sides of a rectangle are equal in length, the perimeter \(P\) can be found using the formula: \[P=2l+2w\]

In the same example, with a length of 5 units and a width of 3 units, the perimeter would be

\[2(5)+2(3) = 10+6 = 16\,units \]

In summary:

• The area gives the total space enclosed by the rectangle and is measured in square units (e.g., square centimeters, square meters, square inches).
• The perimeter gives the total distance around the rectangle and is measured in linear units (e.g., centimeters, meters, inches).