Rectangle

A rectangle is a two-dimensional geometrical shape that has four sides, four vertices, and four angles. The opposite sides of a rectangle are equal in length and parallel to each other. Also, each of the four internal angles of a rectangle measures 90°.

Note that a rectangle is a quadrilateral. Some other types of quadrilaterals are:

a. Square
b. Parallelogram
c. Kite
d. Rhombus
e. Trapezoid

When it comes to solving questions related to quadrilaterals, squares and rectangles are easier to deal with than the rest of the others.

Properties Of Rectangle

Let us now look into some of the basic rectangle properties:

a. A rectangle is a quadrilateral with four equal internal angles.
b. Each internal angle of a rectangle measures 90°.
c. As the opposite angles of a rectangle are equal, a rectangle is also a parallelogram.
d. The opposite sides of a rectangle are equal and parallel.
e. The diagonals of a rectangle bisect each other and are of the same length.
f. The two diagonals of a rectangle bisect each other at different angles – one obtuse angle and the other an acute angle.

g. A rectangle whose two diagonals bisect each other at right angles is called a square.
h. As the two equal diagonals of a rectangle bisect each other, the four vertices of a rectangle are equidistant from the point of bisection. This means a circumcircle can be formed with its center at the point of bisection of the diagonals and its circumference passing through the four vertices of the rectangle. The diameter of the circumcircle is equal to the diagonal of the rectangle.

Formulas Of Rectangle

Some of the important rectangle formulas are as under:

i. Area of a Rectangle, A = a × b, where a and b are the length and the breadth of the rectangle, respectively. That is:

ii. Perimeter of a Rectangle, S = Total length of all sides of the rectangle = 2 (a + b).

iii. Each of the diagonals of a rectangle divides the rectangle into two right-angled triangles with the diagonal being the hypotenuse.

As per Pythagoras’ Theorem, the square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides.

So, we have, (Diagonal)² = (Length)² + (Breadth)².

From this,

:

Example 1: If the length and the breadth of a rectangle are 8 cm and 5 cm respectively, find its area and perimeter.

Solution:

Here, length of the rectangle, a = 8 cm
Breadth of the rectangle, b = 5 cm.

Therefore, area = Length (a) × Breadth (b) = (8 × 5) cm² = 40 cm².
Perimeter = 2 (Length + Breadth) = 2 (8 + 5) = 2 × 13 = 26 cm.

Rhombus

 A rhombus (plural rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhombus is often called a diamond, after the diamonds suit in playing cards which resembles the projection of an octahedral diamond, or a lozenge, though the former sometimes refers specifically to a rhombus with a 60° angle (see Polyiamond), and the latter sometimes refers specifically to a rhombus with a 45° angle.

Properties of a Rhombus

• All sides are congruent.
• Opposite angles are congruent.
• The diagonals are perpendicular to and bisect each other.
• Adjacent angles are supplementary (For eg., ∠A + ∠B = 180°).
• A rhombus is a parallelogram whose diagonals are perpendicular to each other.

Important formulas for a Rhombus

If a and b are the lengths of the diagonals of a rhombus,

• Area = (a* b) / 2
• Perimeter = 4L