# Rectangle (Geometry) : Definition & Properties

## What is Rectangle?

Rectangle is a quadrilateral in which opposite sides are equal and all angles measure 90 degrees

Keyword for Rectangle
(a) Opposite sides are equal
(b) All angles measure 90 degree

## Structure of Rectangle

Rectangle consists of following components

(a) Length
The longest side in the rectangle is called Length
Here AB & CD are the lengths of rectangle

(b) Breadth
The shorter side is called Breadth
here BC & AD are the breadths of rectangle

(c) Vertex
The point where two sides join is called Vertex
A Rectangle has four vertex A, B, C & D

(d) Angles
There are four angles in rectangle.
All angle measures 90 degree.
∠A = ∠B = ∠C = ∠D = 90 degree

## Properties of Rectangle

(a) Opposite sides are equal

Given above is the rectangle ABCD
Here,
AB = CD = 3 cm
AD = BC = 2 cm

(b) All interior angle measures 90 degree

In the above rectangle ABCD, all angle measure 90 degrees
∠A = ∠B = ∠C = ∠D = 90 degree

(c) Both the diagonals of rectangle have same length

AC & BD are the diagonals.
According to Rectangle Property: AC = BD

(d) In Rectangle, diagonals bisect each other

When lines bisect each other it means that the lines divides into equal halves.
Hence,
AO = OC
DO = OB

(e) Length of Diagonal of Rectangle

Formula for length of diagonal of rectangle is:
d\ =\sqrt{a^{2} +b^{2}}

Proof

Given above is the triangle ABCD with diagonal BD
Length of rectangle = a cm
Breadth of Rectangle = b cm

Observe triangle BCD is a right triangle

Applying Pythagoras Theorem
d^{2} =\ \ a^{2} +b^{2} \\\ \\ d\ =\sqrt{a^{2} +b^{2}}

Hence Proved

(f) Diagonals of rectangle bisect at different angles

In the intersection of diagonals you will find two angle of intersection. One angle will be acute and other will be obtuse.

Given above is rectangle PQRS with diagonals PR & QS intersecting at point O.
The two intersection angles are 52 and 128 degree

(f) Perimeter of Rectangle

Perimeter is the total length of the boundary.

In Rectangle;
Perimeter = Length + Breadth + Length + Breadth
Perimeter = 2 x ( Length + Breadth)

(g) Area of Rectangle

The region covered by the figure is called Area

In Rectangle;
Area = Length x Breadth

(h) Rectangle’s line of symmetry

Line of symmetry is a line which divides the geometrical figure into two equal halves.

In Rectangle, there are two lines of symmetry

(i) Rectangle Circumcircle

A circle which touches all the vertex of geometrical figure is called circumcircle.

In rectangle, the center of the circumcircle is located at the intersection of diagonals

Given above is rectangle ABCD with diagonal AC & BD.

The diagonals intersected at point O which is also the center for the circumcenter.

From point O, you can easily draw circle which will touch all the vertex of rectangle.

Radius of Circumcenter
Radius of circumcenter = half of diagonal
Radius of circumcenter = ( d / 2 )

## Frequently asked Questions – Rectangle

(01) How is Square and Rectangle are different?

(a) In square all sides are equal while in Rectangle opposite sides are equal.
(b) In square, diagonals intersect at 90 degrees while in rectangle diagonals intersect at different angles

(02) Is Rectangle a parallelogram?

Properties of Parallelogram are:
(a) Opposite sides are parallel & equal
(b) Opposite angles are equal

Both the condition applies for rectangle.
Hence rectangle is form of parallelogram

(03) How are rectangle and parallelogram different?

In rectangle, all angles are 90 degree.
The same is not the case with parallelogram

(04) Any real life example of rectangle?

## Questions on Rectangle

(01) Find the area of following rectangle

Solution
Length of Rectangle = 6 cm
Breadth of Rectangle = 3 cm

We know that:
Area of rectangle = Length x Breadth
Area of Rectangle = 6 x 3 sq. cm = 18 sq. cm

Hence, 18 sq. cm is the area of above image

(02) Find the area of rectangle of the length is 10 cm and the breadth is 5 cm

Solution
Length of rectangle = 10 cm
Breadth of rectangle = 5 cm

Area of Rectangle = Length x Breadth
Area of Rectangle = 10 x 5 = 50 sq. cm

Hence, 50 sq. cm is the solution

(03) Find the perimeter of the below rectangle

Solution
Length of rectangle = 5 cm
Breadth of rectangle = 3 cm

Perimeter of rectangle = 2 ( Length + Breadth)
Perimeter of rectangle = 2 (5 + 3) = 16 cm

Hence, 16 cm is the solution

(04) The length and diagonal of rectangle is 4 cm and 5 cm respectively. Find the breadth of the rectangle

We have already discussed the formula for diagonal as:

d^{2} =\ \ length^{2} +breadth^{2}\\\ \\ 5^{2} =\ \ 4^{2} +breadth^{2}\\\ \\ 25\ =\ 16\ +\ breadth^{2}\\\ \\ breadth^{2} \ =\ 9\\\ \\ breadth\ =\ 3\

Hence, breadth of rectangle is 3 cm

(05) A rectangle if formed by joining two squares of side 2 cm. Find the area of rectangle

When two squares of 2 cm are joined it will produce rectangle of length 4 cm and breadth 2 cm

Length of rectangle = 4 cm
Breadth of rectangle = 2 cm

Area of Rectangle = Length x Breadth
Area of Rectangle = 4 x 2 = 8 sq. cm

Hence, 8 sq. cm is the solution

## Summary of Rectangle Properties

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