Rectangle

When you look at an object, some questions come to mind and right there you give quick answers to a number of them. One of such questions answered is the shape of the object.

In this article, we will explore the definition of a rectangle, its properties, the formulas for the perimeter and area of a rectangle, and examples of their application.

Definition of rectangle

A rectangle is a special case of a parallelogram. In other words, what makes a parallelogram become a rectangle is having its sides perpendicular to each other. This can be illustrated in the image below.

Rectangle - StudySmarter Original

We can notice that the opposite sides AB and CD are equal in size and parallel, the same for BC and AD. Moreover, the four sides are perpendicular to each other, thus, the quadrilateral is a rectangle.

Rectangles Properties

A rectangle being a parallelogram has all the properties of a parallelogram, but being a special case of it, it has its own unique properties that make it the geometric shape it is.

To better understand the properties of a rectangle, let's consider the following rectangle ABCD in the image below.

Rectangle ABCD - StudySmarter Original

Construction of a rectangle

For the construction of a rectangle, follow these steps.

Step 1: Draw a straight line (R), then place 2 points A and B on the line.

Step 2: Draw 2 perpendicular lines (S) and (T), passing by the two points A and B.

Step 3: Locate two points C and D respectively on the two lines (S) and (T). However, C and D must be on the same level.

The three steps mentioned earlier can be illustrated in the drawing below:

Image resulting from steps 1, 2 and 3 - StudySmarter Original

Step 4: Draw a straight line joining the two points C and D, as shown in the image below:

Image resulting from step 4 - StudySmarter Original

After reaching step 4, you will notice that the 4 points A, B, C, and D will form a rectangular shape.

Formula for the area of a rectangle

The area of a shape is usually measured considering the number of unit squares that cover the surface of the shape. Square centimeters, square feet, square inches, and other similar units are used to measure area.

Given a rectangle with height h and base b, its area will be equal to:

$A=b\times h$.

Find the area of the rectangular shape in the image below. Consider the square composed of 25 smaller squares the square of side 1 unit.

We can notice that the height of the rectangle is equal to 2 unit squares, so its length is 2 units. Similarly, the base of the rectangle is 5 units. So the area of this rectangle can be calculated by multiplying the height by the base:

A= 2 unit × 5 unit = 10 unit²

Formula for the perimeter of a rectangle

Consequently, the shape's perimeter is calculated by summing the lengths of all its sides. The same concept also applies to a rectangular shape. So, the total length of all the sides of a rectangle is known as the perimeter.

A rectangle has its opposite sides equal to each other (one of its properties). Thus, the rectangle's perimeter of a rectangle with sides of lengths a, b, a, b is P = a + b + a + b, or P = 2a + 2b, or even P = 2 (a + b).

So, we just need to calculate the lengths of two sides to find the perimeter of a rectangle since opposite sides of a rectangle are always equal.

Square and rectangle

You can notice in the figure below that a square and a rectangle are both quadrilateral with four sides.

A rectangle and a square - StudySmarter Original

A square and a rectangle have similar properties as illustrated in the table below:

What Characterizes a Square as a Unique Rectangle?

As illustrated in the table above, a square is a special type of rectangle for the following reasons:

1. A square has all the properties of a rectangle.

2. The only two differences between the square and the rectangle are that a square has its diagonal perpendicular to each other, and all its sides are equal.