What is a Rectangle?
A rectangle is a two dimensional shape. It is a quadrilateral whose opposite sides are parallel and equal in length and all four interior angles measure 90\(^\circ\) each. It is also known as an equiangular quadrilateral.
The longer side of a rectangle is known as the length and the shorter side is called its width.
Furthermore, the diagonals of a rectangle bisect each other.
Properties of Rectangles
The properties of a rectangle are based on its side lengths, angles and diagonals. Let us explore these properties.
Fundamental properties of a rectangle:
- A rectangle has four sides and four vertices.
- All angles of a rectangle measure 90 degrees.
- The opposite sides of a rectangle are equal and parallel.
- The diagonals of a rectangle cut each other in half.
- The perimeter of a rectangle is equal to the sum of the measure of its sides.
- The area of a rectangle is equal to the product of its length and width.
- A rectangle is a four-right-angle parallelogram.
- The sum of all interior angles of a rectangle is 360 degrees.
- The diagonals of a rectangle bisect each other at two different angles- one acute angle and one obtuse angle.
- The rectangle is known as a square if the two diagonals bisect each other at right angles.
- When the rectangle is rotated along the line connecting the midpoints of the longer parallel sides, it forms a cylinder. The height of the cylinder is equal to the width of the rectangle in this case. In addition, the circumference of the circular base or top of the cylinder is equal to the length of a rectangle.
- When the rectangle is rotated along the line connecting the midpoints of the shorter parallel sides, it forms a cylinder. The height of the cylinder is equal to the length of the rectangle in this case. Similarly, the width of a rectangle is equal to the circumference of the circular base or top of the cylinder.
Properties of sides of a rectangle
Let us take rectangle ABCD as the reference rectangle.
Opposite sides are equal AB = DC, AD = BC
Diagonal are equal AC = BD, DO = OB
Diagonal bisect each other AO = OC, DO = OB
Properties of the angles of a rectangle
All interior angles are equal to 90\(^\circ\). That is,
ㄥA = ㄥB = ㄥC = ㄥD = 90\(^\circ\)
Rectangle Formulas
If the length of the rectangle is l and the width is w then,
1. The perimeter of a rectangle is defined as the measure of the boundary of the rectangle. It is measured in the same units as its sides. If its length is l and its width is w, then,
Perimeter of rectangle, P = 2 × (l + w)
2. The amount of space covered by a two-dimensional shape in a plane is called its area. The area is measured in square units. The area of a rectangle equals the product of the length and width.
The formula for calculating the area of a rectangle is:
Area of a rectangle, A = Length × width or l × w
3. A rectangle has two diagonals of equal length that bisect each other. Each diagonal divides the rectangle into 2 right triangles.
In the figure below, diagonal AC divides the rectangle into two right triangles – \(\Delta\)ABC and \(\Delta\)ADC.
Let ‘d’ be the diagonal of the rectangle. The rectangle’s length and width, that is l and w, represent the base and height of the right triangle respectively. The diagonal is the hypotenuse of the right triangle. As a result, the length of the diagonal will be:
Diagonal (d) = \(\sqrt{l^2+w^2}\)
Solved Exampless
Example 1. Find the perimeter of a TV Screen whose sides are 45 inches and 35 inches?
Solution:
Given data,
Length of the TV Screen = 45 inches
Width of the TV Screen = 35 inches
As per the known properties of a rectangle, the formula for calculating the perimeter of a rectangle is:
Perimeter of rectangle = 2 × (l + w)
Now, substituting the values of l and w.
P = 2 × (45 + 35)
P = 2 × (80)
P = 160 inches
Thereby, the perimeter of the TV screen is 160 inches.
Example 2. Find the area of brick having a length of 19 centimeters and a width of 9 centimeters?
Solution:
Data provided,
Length of the brick = 19 centimeters
Width of the brick = 9 centimeters
As per the known properties of a rectangle, the formula for calculating the area of a rectangle is:
Area of rectangle = (l × w)
Now, substituting the values of l and w.
P = (19 × 9)
P = 171
Therefore, the area of a brick is 171 square centimeters.
Example 3. What is the diagonal of a big wall having a length of 14 meters and a width of 1 meter?
Solution:
Given data,
Length of the big wall = 14 meters
Width of the big wall = 1 meter
As per the known properties of a rectangle, the formula for calculating the diagonal length is:
Diagonal (d) of a big wall = \(\sqrt{l^2+w^2}\)
Now, substituting the values of l and w.
D = \(\sqrt{14^2+1^2}\)
D = \(\sqrt{196+1}\)
D = \(\sqrt{197}\)
D = 14.03
Thereby, the diagonal of a big wall is 14.03 meters.
A square is a special type of rectangle because it has some properties in common, which are listed below.
The interior angles of a square and a rectangle are both 90 degrees.
Both shapes have equal and parallel opposite sides.
The length of the diagonals that bisect each other is the same.
In a square all four sides are equal in length. However, in a rectangle only the opposite sides are equal in length.
The properties of a rectangle’s diagonal are as follows:
- The diagonals of a rectangle are equal in length.
- The diagonals meet at two different angles – acute and obtuse.
- The diagonals bisect each other.
- The Pythagorean Theorem can be used to find the length of the diagonals.
- The diagonals divide the rectangle into two congruent right triangles.
- The diagonals are the hypotenuse of these right triangles.