Parallelogram Formula and Rhombus Formula

Parallelogram Formula

A parallelogram is a 2-dimensional shape that has four sides and has two pairs of parallel lines. The parallelogram consist of equal opposite sides and its opposite angles are equal in measure. Let us now discuss the parallelogram formula i.e. area and perimeter of the parallelogram.

Parallelogram Formula

What is a Parallelogram?

A parallelogram is a quadrilateral that has two pairs of parallel and equal sides. A quadrilateral is said to be a parallelogram if the two pairs of opposite sides in a quadrilateral are equal, then it is a parallelogram.

If two opposite sides in a quadrilateral are parallel and equal, then this quadrilateral is a parallelogram; if, in a quadrilateral, the diagonals bisect each other, then this quadrilateral is a parallelogram.

Properties of Parallelogram

There are some important properties of parallelograms:

1. In a parallelogram opposite sides are equal
2. In a parallelogram opposite angels are equal.
3. The sum of adjacent angles are supplementary i.e. (∠B + ∠C = 180°).
4. In a parallelogram, if one angle is right, then all angles are right.
5. Diagonals of a parallelogram bisect each other.
6. In a parallelogram, each diagonal of a parallelogram divides it into two congruent triangles.

The Perimeter of a Parallelogram

The perimeter is the sum of the length of all the 4 sides. In a parallelogram opposite sides are equal so the perimeter is:

P = 2b + 2a

P = 2( a + b)

Where,

a	Length of the side of the parallelogram
b	Length of the side of the parallelogram

Area of Parallelogram

The area of a parallelogram is the number of square units inside the polygon. The area of a parallelogram can be determined by multiplying the base and height. For finding the area of a parallelogram, the base and height must be perpendicular. The area of a parallelogram is given by

Area =b × h

Where,

b	Base of Parallelogram
h	The altitude of the parallelogram

Derivation of Area of Parallelogram

Let ABCD be the parallelogram whose area is being derived. We extend AB to F such as F as the foot of the altitude from C. Now, construct the point E such that DE is the altitude from D.

In ΔAEDANDΔBFC

AD = BC

∠AED=∠BFC

DE = CF

Therefore,

ΔAED≅ΔBFC

area (ΔAED)=area(ΔBFC)

So,

Area (ABCD) = EF  ͯ FC = AB  ͯ CF

Solved Examples

Q.1: Find the area of a parallelogram with a base is 10 cm and a height is 5 cm.

Solution: Given, Base = 10 cm, Height = 5 cm

Area of Parallelogram = B × H

Area = (10 cm) ͯ× (5 cm)

Area = 50 cm2

Therefore, Area of Parallelogram is 60 cm2.

Q.2: Find the perimeter of a parallelogram whose base is 20 cm and height 12 cm?

Solution: Given,
Base = 20 cm
Height = 12 cm

Perimeter of a Parallelogram = 2(Base + Height)
= 2(20 + 12)
= 2 × 32 cm
= 64 cm

Therefore, Perimeter of a Parallelogram is 64 cm

Rhombus Formula

A rhombus is a 2-dimensional shape that has four equal sides. Rhombus consists of all sides equal and its opposite angles are equal in measure. Let us now discuss the rhombus formula i.e. area and perimeter of the rhombus.

Rhombus Formula

What is a Rhombus?

Rhombus is a special type of parallelogram that has all sides equal. Rhombus is quadrilateral whose all sides are equal.

Properties of Rhombus

1. In a rhombus all sides are equal
2. In a rhombus opposite angles are equal.
3. Also, in a rhombus the sum of adjacent angles are supplementary i.e. (∠B + ∠C = 180°).
4. In a rhombus, if one angle is right, then all angles are right.
5. In a rhombus, each diagonal of a rhombus divides it into two congruent triangles.
6. Diagonals of a rhombus bisect each other and also perpendicular to each other.

The Perimeter of a Rhombus

The perimeter is the sum of the length of all the 4 sides. In rhombus all sides are equal.

So, Perimeter of rhombus = 4 × side 

P = 4s

Where,

s	length of a side of a rhombus

Area of Rhombus

The area of a rhombus is the number of square units inside the polygon. The area of a rhombus can be determined in two ways:

i) By multiplying the base and height as rhombus is a special type of parallelogram.

Area of rhombus = b  × h

Where,

b	Base of Rhombus
h	Height of the Rhombus

ii) By finding the product of the diagonal of the rhombus and divide the product by 2.

Area of rhombus = 12  × d₁  × d₂

where,

d₁, d₂

Diagonals of Rhombus

Derivation of Area of Rhombus

Let ABCD is a rhombus whose base AB = b, DB ⊥ AC,   DB is diagonal of rhombus = d₁, AC is diagonal of rhombus = d₂, and the altitude from C on AB is CE, i.e., h.

i) Area of rhombus ABCD   = 2 Area of ∆ ABC

= 2  ×  12 AB  × CD sq units.

= 2  ×  12 b  × h sq. units

= base × height sq. units

ii) Area of rhombus    = 4  × area of ∆ AOB

= 4  ×  12  ×  AO  ×  OB sq. units

= 4  ×  12  × 12 d₂  ×  12 d₁ sq. units

so,

= 4  ×  18 d₁  ×  d₂ square units

= 12  ×  d₁  × d₂

Therefore, area of rhombus = 12(product of diagonals) square units

Solved Examples

Q.1. What is the perimeter of a rhombus ABCD whose diagonals are 16 cm and 30 cm ?

Solution: Given d1 = 30 cm and d2 = 16 cm
AO= 302=15cm,

BO= 162=8cm,

∠AOB=90∘

From Pythagorean Theorem, we know
AB2=AO2+BO2
AB =289−−−√

=17 cm

Since, AB=BC=CD=DA,
Perimeter of ABCD = 17 × 4 = 68 cm

Q.2. Find the area of the rhombus having each side equal to 17 cm and one of its diagonals equal to 16 cm.

Solution: In rhombus ABCD, AB = BC = CD = DA = 17 cm
AC = 16 cm, AO = 8 cm

In ∆ AOD,

AD² = AO² + OD²

17² = 8² + OD²

289 = 64 + OD²

225 = OD²

OD = 15 cm

Therefore, BD   = 2 OD

= 2  × 15

= 30 cm

Now, area of rhombus = 12  ×  d₁  × d₂

= 12× 16  × 30

= 240 cm²