Rectangle

A rectangle is a quadrilateral in which all the angles are equal and the opposite sides are equal and parallel. There are many rectangular objects around us. Each rectangle shape is characterized by two dimensions, its length and width. The longer side of the rectangle is known as the length and the shorter side is known as the width. In this chapter, let us learn about the rectangle shape, some rectangle formulas, types of rectangles and its properties.

A rectangle is a closed two-dimensional figure with four sides. The opposite sides of a rectangle are equal and parallel to each other and all the angles of a rectangle are equal to 90°. Observe the rectangle given below to see its shape, sides and angles.

Definition of a Rectangle

The definition of a rectangle states that a rectangle is a 2D shape with 4 sides, 4 vertices and 4 interior right angles. The opposite sides of a rectangle are equal and parallel.

Length and Breadth of a Rectangle

The length of a rectangle refers to the longer side and the breadth of a rectangle refers to the shorter side. It should be noted that the length of a rectangle can be calculated if we know the area, perimeter and the breadth of the rectangle. Similarly, the breadth of a rectangle can be calculated if we know the area, perimeter and the length of the rectangle.

A rectangle is a closed figure which has four sides and the angle formed by adjacent sides is 90°. A rectangle can have a wide range of properties. Some of the important properties of a rectangle are given below.

• A rectangle is a quadrilateral.
• The opposite sides of a rectangle are equal and parallel to each other.
• The interior angle of a rectangle at each vertex is 90°.
• The sum of all interior angles is 360°.
• The diagonals bisect each other.
• The length of the diagonals is equal.
• The length of the diagonals can be obtained using the Pythagoras theorem. The length of the diagonal with sides a and b is, diagonal = √( a² + b²).
• Since the sides of a rectangle are parallel, it is also called a parallelogram.
• All rectangles are parallelograms but all parallelograms are not rectangles.

Rectangle Formulas

A rectangle has a few basic formulas which can be noted in order to find the missing or unknown values. A few rectangle formulas are given below:

• Area of rectangle formula: If we know the length and breadth of a rectangle, we can find the area using the formula, Area of rectangle = Length × Breadth
• Perimeter of rectangle formula: If we know the length and breadth of a rectangle, we can find the perimeter using the formula, Perimeter of rectangle = 2 (Length + Breadth)
• Length of rectangle formula: If we know the area and breadth of a rectangle, we can find the length using the area formula by substituting the values or we can reframe the area formula as, Length = Area of rectangle ÷ Breadth. Similarly, if we know the perimeter and breadth of a rectangle, we can find the length using the perimeter formula by substituting the values or we can reframe the perimeter formula as, Length = (Perimeter ÷ 2) - Breadth
• Breadth of rectangle formula: If we know the area and length of a rectangle, we can find the breadth using the area formula by substituting the values or we can reframe the area formula as, Breadth = Area of rectangle ÷ Length. Similarly, if we know the perimeter and length of a rectangle, we can find the breadth using the perimeter formula by substituting the values or we can reframe the perimeter formula as, Breadth = (Perimeter ÷ 2) - Length

The diagonal of a rectangle is a line segment that joins any two of its non-adjacent vertices. In the following rectangle, AC and BD are the diagonals that are equal in length. A diagonal cuts a rectangle into 2 right-angled triangles, in which the diagonal forms the hypotenuse, and the two adjacent sides of the rectangle form the other two sides of the triangle.

Diagonal of Rectangle Formula

The formula for the diagonal of a rectangle is derived using the Pythagoras theorem. Following the figure given above, let us consider a rectangle of length 'l' and width 'w'. Let the length of each diagonal be 'd'. Applying Pythagoras theorem to the triangle ABD, d² = l² + w². Taking square root on both sides, √(d²) = √( l² + w²). Thus, the diagonal of a rectangle formula is diagonal (d): √(l² + w²) and thus the diagonals of a rectangle can be calculated when the length and width of the rectangle are known.

Diagonal of Rectangle (d) = √(l² + w²)

The area of a rectangle is the number of unit squares that can fit into a rectangle. In other words, the space occupied by a rectangle is the area of the rectangle. Some examples of rectangular shapes are the flat surfaces of laptop monitors, blackboards, painting canvas, etc. We can use the formula of the area of a rectangle to find the space occupied by these objects. For example, let us consider a rectangle of length 4 inches and width 3 inches. Let us draw unit squares inside the rectangle. Each unit square is a square of length 1 inch. Now, count the number of unit squares in the below figure. How many squares can you observe? There are 12 squares in all. We have already learned that area is measured in square units. Since the unit of this rectangle is given in inches, the area is measured and written in square inches. Thus, the Area of the rectangle = 12 square inches. Thus, the area of a rectangle can be calculated when its sides (length and width) are known.

Area of Rectangle Formula

The formula for the area of a rectangle whose length and breadth (width) are 'l' and 'w' respectively is the product of its length and width, that is:

Area of a Rectangle = (l × w)

The perimeter of a rectangle is the length of the total boundary of a rectangle. It can be taken as the sum of the total measure of the length and width of the rectangle and it is expressed in linear units like cm, inches, and so on. For example, if you need to decorate the border of your rectangular notebook, you can easily calculate how much ribbon you would need by finding the perimeter or if you need to put a fence around your garden, the perimeter of the garden will give you the exact length of wire you would need. The formula used to calculate the perimeter of a rectangle is explained below.

Perimeter of Rectangle Formula

The formula for the perimeter, 'P' of a rectangle whose length and (breadth) width are 'l' and 'w' respectively is 2(l + w).

Perimeter of a Rectangle Formula = 2 (Length + Width)

Types of Rectangles

A quadrilateral whose opposite sides are equal and adjacent sides meets at 90° is called a rectangle. A rectangle has two equal diagonals. The length of the diagonals is calculated by using the length and width. There are two types of rectangles:

• Square
• Golden Rectangle

Square

A square is a closed two-dimensional shape with four equal sides and four equal angles. It is a type of rectangle in which all four sides are equal. The interior angle at each vertex is 90º which satisfies the definition of the rectangle. Observe the square given below which fulfills all the properties of being a rectangle.

Golden Rectangle

A golden rectangle is a rectangle whose 'length to the width' ratio is similar to the golden ratio, 1: (1+⎷5)/2. Its sides are defined according to the golden ratio, that is, 1: 1.618. For instance, if the width is about 1 foot long then the length will be 1.168 feet long.

☛ Related Articles

• Properties of Rectangle
• Difference Between Square and Rectangle
• Is a square a rectangle?

What is a Rectangle in Geometry?

A rectangle is a two-dimensional shape (2D shape) in which the opposite sides are parallel and equal to each other and all four angles are right angles. The shorter side of a rectangle is called its breadth (width) and the longer side is the length. One of the most common geometrical figures that we see in our day-to-day life is a rectangle.

☛ Also Read

• 3D Shapes
• Geometric Shapes

How to Find the Length of a Rectangle?

• The length of a rectangle can be calculated if the perimeter and the width is known. For example, if the perimeter of a rectangle is 32 units and its width is 4 units, we can use the perimeter formula to get the length. Perimeter of a rectangle = 2 (l + w). Let us substitute the known values in the formula, 32 = 2 (length + 4). On solving this, we get length + 4 = 16, that is, length = 12 units. Similarly, if the area of the rectangle and the width is known, the length can be calculated by using the formula for the area of a rectangle and substituting the known values.
• Another way to find the length of a rectangle is by using the following formulas that have been derived from the area and perimeter formulas, Length = Area of rectangle ÷ Width; and Length = (Perimeter ÷ 2) - width

How to Find the Width of a Rectangle?

• The width of a rectangle can be calculated if the area and the length is known. For example, if the area of a rectangle is 48 units and its length is 12 units, we can use the area formula to get the width. Area of a rectangle = l × w. Let us substitute the known values in the formula, 48 = 12 × width. On solving this, we get width = 48/12, that is, width = 4 units. Similarly, if the perimeter of the rectangle and the length is known, the width can be calculated by using the formula for the perimeter of a rectangle and substituting the known values.
• Another way to find the width of a rectangle is by using the following formulas that have been derived from the area and perimeter formulas, Width = Area of rectangle ÷ Length; and Width = (Perimeter ÷ 2) - Length

How to Find the Diagonal of a Rectangle?

The length of the diagonal of a rectangle can be calculated if the length and width is known. Since the diagonal of a rectangle forms a right-angled triangle with its sides, the diagonal becomes the hypotenuse and its value can be calculated using the Pythagoras theorem. For example, if the length of a rectangle is given as 4 units and the width is 3 units, we can find the length of the diagonal using the formula, diagonal (d) = √(l² + w²). So, let us substitute the value of the length and width in it. Diagonal (d) = √(l² + w²) = √(4² + 3²) = √(16 + 9) = √25 = 5 units.

Why is a Rectangle Not a Regular Polygon?

A regular polygon is a polygon in which all the sides are of equal length and all the angles are of equal measure. In a rectangle, only the opposite sides are of equal length, therefore, a rectangle is not a regular polygon.

☛ Also Check:

• Types of Polygon
• Area of Polygons

What is the Formula for the Area of a Rectangle?

The area of a rectangle is the space occupied by it. The area of a rectangle is the product of its length and breadth (width). The formula for calculating the area of a rectangle is: Area = L × B; where (L) is the length and (B) is the breadth (width) of the rectangle.

☛ Check out the list of formulas for easy calculations:

• Perimeter Formulas
• Volume Formulas
• Surface Area Formulas
• Measurement Formulas

What are the Properties of a Rectangle?

The basic properties of a rectangle are as follows:

• The opposite sides are parallel and equal.
• All the angles are equal to 90°.
• The diagonals are equal and they bisect each other.

Is the Area of the Rectangle the Same as the Area of the Square?

No, the area of the square is not necessarily the same as the area of the rectangle because every square is a rectangle but all rectangles are not squares. The formula to calculate the area of a rectangle is, Area of rectangle = Length × Width, and area of a square = (side)².

How to Find the Perimeter of a Rectangle?

The perimeter of a rectangle is twice the sum of its length and width and it is expressed by the formula, Perimeter = 2 (Length + Width). It is expressed in linear units like cm, inches, and so on.

What is a 3D Rectangle Called?

A 3D rectangle is called a rectangular prism. A rectangular prism is a prism whose bases are also rectangles. It has 6 faces in all, out of which there are 3 pairs of identical opposite faces, i.e., all the opposite faces are identical in a rectangular prism.

What does a Rectangle Look Like?

A rectangle looks like an elongated square in which 2 opposite sides are equal and parallel. A few common examples of rectangles can be seen in the screen of a laptop, a notebook, a mobile phone and so on.

What is Length and Breadth in Rectangle?

The length of a rectangle refers to the longer side and the breadth of a rectangle refers to the shorter side.

• The length of a rectangle can be calculated using the following formulas that have been derived from the area and perimeter formulas, Length = Area of rectangle ÷ Width; and Length = (Perimeter ÷ 2) - width.
• Similarly, the width of a rectangle can be calculated by using the following formulas that have been derived from the area and perimeter formulas, Width = Area of rectangle ÷ Length; and Width = (Perimeter ÷ 2) - Length