Properties of Rectangle

Introduction

In daily life, we come across various geometrical objects, which are bounded by straight line segments. These are known as polygons. The polygons are categorized into various types based on their number of sides and angles. The rectangle is one type of polygon extensively used in Euclidean geometry. Various objects such as black boards, carrom boards, books, doors, smartphones, beds, etc. These are real-life examples of the rectangle. In this tutorial, we will discuss the meaning, properties, types, and basic formulae related to the rectangle with solved examples.

Rectangle

A rectangle is a two-dimensional polygon embedded in four straight line segments. In other words, it is defined as a quadrilateral having four sides and angles. It is also an equiangular quadrilateral with two parallel pairs of equal sides. In addition, the measurement of each angle is 90°. The term “rectangle” originated from the Latin words “rectus” (means right or proper) and “ angulus ” (means angle). The longer side of the rectangle is known as the length, and the smaller side is known as the width of the rectangle. Moreover, it has two diagonals such as PR and QS. For illustration purposes, a rectangle PQRS is represented below.

Properties of Rectangle

The properties help us to identify and recognize the rectangle from other geometrical constructions. There are several properties of the rectangle that are briefly summarized below.

• The rectangle is a quadrilateral having four sides and angles.

• The opposite sides are parallel and equal to each other.

• The measure of each angle is 90°.

• The addition of interior angles of a rectangle is 360°.

• The length of the diagonals is equal.

• The diagonals bisect each other.

• The rectangle can be called a parallelogram.

• The length of the diagonal can be evaluated using the Pythagoras theorem. If the measure of length and width are p and q, respectively, the length of the diagonal will be $\mathrm{\sqrt{p^2+q^2}}$.

• A rectangle is said to be a square if the diagonals are orthogonal.

Formulae of Rectangles

The formulae of rectangle help to find the area and perimeter. Let’s consider a rectangle with a length and width of p and q, respectively. Now, the area and perimeter of a rectangle can be determined using the following formulae.

• Area of the rectangle = p×q

• perimeter of the rectangle = 2(p+q)

Rectangles in Real life

We see various real-life examples of rectangles in our daily lives. Some of them are mentioned below.

Examples	Descriptions	Images
Laptop	It looks like a quadrilateral having equal parallel opposite sides.
Books	Although books are cuboidal shapes; however each face of the book is a rectangle.
Smart phones	The top and bottom faces of the smartphone are rectangular shapes.
Doors	The doors of the house are common examples of the rectangle.
Chocolate	The bar-shaped chocolates are rectangular shapes.
Table top	The dining tables or study tables are rectangular shapes.
Bed	A single bed is an example of a rectangle.
Black board	The black board has a rectangular face.
Ruler	The geometrical tool has rectangular structures.
Envelope	It has a flat rectangular shape.
Currency notes	The currency notes are also of rectangular shapes.
Debit or credit cards	The shape of the plastic cards is rectangles.

Solved Examples

Example 1

The measure of length and width of a rectangular blackboard are 4 m and 2 m, respectively. Evaluate the perimeter and area of the blackboard.

Solution:

According to the questions,

The length of the rectangle = p = 4 m

The width of the rectangle = q = 2 m

The perimeter can be determined using the formula

Perimeter = 2(p+q)=2(4+2)=12 m

The area of the rectangle can be obtained using the following formula

Area = p×q=4×2=8 m²

∴ The perimeter and area of the rectangle are 12 m and 8 m², respectively.

Example 2

The perimeter of one rectangular tile is 50 cm. The length of the tile is 15 cm. Find the area of 20 tiles.

Solution:

It is given that,

The perimeter of one rectangular tile is = 50 cm

The length of the tile is = 15 cm

Let’s assume the measure of the width of the tile = q cm

Using the perimeter formula of a rectangle,

Perimeter = 2(length+width)

$$\mathrm{\Rightarrow 50=2(15+q)}$$

$$\mathrm{\Rightarrow 5+q=\frac{50}{2}=25}$$

⇒q=25-15=10 cm

The area of one tile can be determined using the following formula

Area =length×width=15×10=150 cm²

The area of 20 tiles = 150×20=3000 cm²

∴ The area of 20 rectangular tiles is 3000 cm².

Example 3

The length of the diagonal and width of a rectangle are 25 cm and 24 cm, respectively.

Find the area and perimeter of the rectangle.

Solution:

According to the questions,

The measure of the length = 24 cm

The measure of the diagonal = 25 cm

Let’s consider the width of the rectangle = q cm

Using the Pythagoras theorem,

Diagonal²=length²+width²

$$\mathrm{\Rightarrow 25^2=24^2+q^2}$$

$$\mathrm{\Rightarrow q^2=25^2-24^2}$$

⇒q²=625-576=49

$$\mathrm{\Rightarrow q=\sqrt{49}=7 cm}$$

The perimeter of the rectangle is =2(24+7)=62 cm

The area of the rectangle is = 24×7=168 cm²

∴ The area and perimeter of the rectangle are 168 cm² and 62 cm, respectively.

Word Problems

Problem 1: The area and the length of a rectangular tile are 150 〖cm〗^2 and 10 cm. Evaluate the length of the diagonal. Also, find the perimeter of the rectangle.

Problem 2: The perimeter of one rectangular box is 30 cm. The length of the box is 8 cm. Find the area of 100 boxes.

Conclusion

The present tutorial gives a brief introduction about an important polygon, i.e., rectangle. In addition, the basic properties and some formulae related to the perimeter, area, and diagonals of the rectangle have been illustrated. Moreover, some solved examples have been provided for better clarity of this concept. In conclusion, the present tutorial may be useful for understanding the basic concept of the properties of a rectangle.

FAQs

1. Do the diagonals of a rectangle bisect each other orthogonally?

The diagonals of a rectangle bisect each other. However, they don't need to intersect each other at 90°. If they bisect perpendicularly, then the polygon is known as a square.

2. Can a rectangle be a parallelogram?

Yes. All the rectangles are parallelograms since the opposite sides are parallel. However, the reverse is not true.

3. What are the types of the rectangle used in geometry?

There are three more types of the rectangle used in Euclidean geometry: spherical rectangle, elliptic rectangle, and hyperbolic rectangle.

4. What is the addition of the interior and the corresponding exterior angle of a rectangle?

The measure of the interior angle is = 90°

The measure of the corresponding exterior angle is = 90°

The summation of the interior and the corresponding exterior angle of a rectangle is = 90°+90°=180°

5. Name the geometrical structures other than a rectangle?

There are various geometrical structures studied in mathematics, including square, parallelogram, rhombus, trapezium, pentagon, hexagon, octagon, etc.