What is the Area of Parallelogram Rstu

Expanse of a parallelogram
is a region covered by a parallelogram in a 2-dimensional plane. In Geometry, a parallelogram is a ii-dimensional figure with four sides. It is a special instance of the quadrilateral, where opposite sides are equal and parallel. The area of a parallelogram is the space enclosed within its four sides. Area is equal to the production of length and peak of the parallelogram.

The sum of the interior angles in a quadrilateral is 360 degrees. A parallelogram has two pairs of parallel sides with equal measures. Since it is a two-dimensional figure, it has an surface area and perimeter. In this article, let us discuss the surface area of a parallelogram with its formula, derivations, and more than solved problems in detail.

Also bank check:Mathematics Solutions

What is the Area of Parallelogram?

The
expanse of a parallelogram
is the region bounded past the parallelogram in a given two-dimension space. To recall, a parallelogram is a special blazon of quadrilateral which has four sides and the pair of reverse sides are parallel. In a parallelogram, the opposite sides are of equal length and contrary angles are of equal measures. Since the rectangle and the parallelogram have similar properties, the area of the rectangle is equal to the area of a parallelogram.

Expanse of Parallelogram Formula

To notice the surface area of the parallelogram, multiply the base of operations of the perpendicular by its height. Information technology should exist noted that the base and the summit of the parallelogram are perpendicular to each other, whereas the lateral side of the parallelogram is not perpendicular to the base of operations. Thus, a dotted line is drawn to represent the elevation.

Therefore,

Where “b” is the base and “h” is the height of the parallelogram.

Let united states of america learn the derivation of area of a parallelogram, in the next department.

How to Calculate the Expanse of Parallelogram?

The parallelogram area can be calculated, using its base and height. Apart from information technology, the area of a parallelogram can also exist evaluated, if its two diagonals are known along with whatever of their intersecting angles, or if the length of the parallel sides is known, along with any of the angles between the sides. Hence, there are three method to derive the expanse of parallelogram:

• When base and height of parallelogram are given
• When superlative is not given
• When diagonals are given

Area of Parallelogram Using Sides

Suppose a and b are the set of parallel sides of a parallelogram and h is the elevation, then based on the length of sides and elevation of information technology, the formula for its expanse is given by:

Area = Base of operations × Height

A = b × h     [sq.unit of measurement]

Example:
If the base of a parallelogram is equal to 5 cm and the superlative is 3 cm, then notice its area.

Solution: Given, length of base=5 cm and height = 3 cm

As per the formula, Area = 5 × 3 = 15 sq.cm

Expanse of Parallelogram Without Height

If the acme of the parallelogram is unknown to united states, then we tin utilise the trigonometry concept hither to detect its area.

Expanse = ab sin (10)

Where a and b are the length of parallel sides and ten is the angle betwixt the sides of the parallelogram.

Example: The angle between any 2 sides of a parallelogram is ninety degrees. If the length of the two parallel sides is 3 cm and 4 cm respectively, then find the area.

Solution: Let a = iii cm and b=iv cm

x = 90 degrees

Expanse = ab sin (x)

A = 3 × 4 sin (xc)

A = 12 sin xc

A = 12 × 1 = 12 sq.cm.

Notation:
If the angle betwixt the sides of a parallelogram is 90 degrees, then information technology is a rectangle.

Area of Parallelogram Using Diagonals

The area of any parallelogram can also be calculated using its diagonal lengths. Every bit we know, at that place are two diagonals for a parallelogram, which intersects each other. Suppose, the diagonals intersect each other at an angle y, then the area of the parallelogram is given by:

Area = ½ × d_ane
× d_two
sin (y)

Cheque the table below to go summarised formulas of an expanse of a parallelogram.

Where,

• b = base of the parallelogram (AB)
• h = height of the parallelogram
• a = side of the parallelogram (AD)
• x = whatever angle between the sides of the parallelogram (∠DAB or ∠ADC)
• d_i
  = diagonal of the parallelogram (p)
• d₂
  = diagonal of the parallelogram (q)
• y = any angle between at the intersection point of the diagonals (∠DOA or ∠DOC)

Note:
In the above effigy,

• DC = AB = b
• AD = BC = a
• ∠DAB = ∠DCB
• ∠ADC = ∠ABC
• O is the intersecting betoken of the diagonals
• ∠DOA = ∠COB
• ∠DOC = ∠AOB

Surface area of Parallelogram in Vector Form

If the sides of a parallelogram are given in vector form then the area of the parallelogram can be calculated using its diagonals. Suppose, vector ‘a’ and vector ‘b’ are the two sides of a parallelogram, such that the resulting vector is the diagonal of parallelogram.

Surface area of parallelogram in vector form = Modernistic of cross-product of vector a and vector b

A = | a × b|

Now, we take to find the area of a parallelogram with respect to diagonals, say d₁
and d₂,  in vector class.

So, nosotros can write;

a + b = d₁

b + (-a) = d₂

or

b – a = d₂

Thus,

d_ane × d_ii
= (a + b) × (b – a)

= a × (b – a) + b × (b – a)

= a × b – a × a + b × b – b × a

= a × b – 0 + 0 – b × a

= a × b – b × a

Since,

a × b = – b × a

Therefore,

d₁ × d_two
= a × b + a × b = 2 (a × b)

a × b = 1/ii (d₁ × d₂)

Hence,

Surface area of parallelogram when diagonals are given in the vector form, becomes:

A = 1/2 (d₁ × d₂)

where d1 and d2 are vectors of diagonals.

Example: Notice the area of parallelogram whose adjacent sides are given in vectors.

A = 3i + 2j and B = -3i + 1j

Expanse of parallelogram = |A × B|

\(\begin{array}{l}= \begin{vmatrix} i & j & k\\ 3 & 2 & 0\\ -three & 1 & 0 \end{vmatrix}\end{array} \)

= i (0-0) – (0-0) + chiliad(three+6)  [Using determinant of 3 x 3 matrix formula]

= 9k

Thus, area of the parallelogram formed by two vectors A and B is equal to 9k sq.unit.

Related Articles

Solved Examples on Area of Parallelogram

Question i: Find the area of the parallelogram with the base of 4 cm and acme of 5 cm.

Solution:

Given:

Base of operations, b = 4 cm

h = 5 cm

We know that,

Area of Parallelogram = b×h Square units

= iv × 5 = 20 sq.cm

Therefore, the area of a parallelogram = 20 cm²

Question ii: Notice the surface area of a parallelogram whose latitude is 8 cm and height is 11 cm.

Solution:

Given,

b = eight cm

h = 11 cm

Area of a parallelogram

= b × h

= eight × 11 cm²

= 88 cm²

Question 3: The base of the parallelogram is thrice its meridian. If the expanse is 192 cm², notice the base and superlative.

Solution:

Permit the meridian of the parallelogram = h cm

then, the base of the parallelogram = 3h cm

Area of the parallelogram = 192 cm²

Area of parallelogram = base × height

Therefore, 192 = 3h × h

⇒ three × h²
= 192

⇒ h²
= 64

⇒ h = 8 cm

Hence, the height of the parallelogram is eight cm, and latitude is

3 × h

= 3 × eight

= 24 cm

Discussion Problem on Area of Parallelogram

Question: The area of a parallelogram is 500 sq.cm. Its height is twice its base. Detect the height and base.

Solution:

Given, area = 500 sq.cm.

Tiptop = Twice of base of operations

h = 2b

By the formula, nosotros know,

Area = b 10 h

500 = b 10 2b

2bⁱⁱ
= 500

b²
= 250

b = 15.8 cm

Hence, height = ii x b = 31.half-dozen cm

Exercise Questions on Surface area of a Parallelogram

1. Find the surface area of a parallelogram whose base is 8 cm and height is 4 cm.
2. Find the area of a parallelogram with a base equal to seven inches and height is 9 inches.
3. The base of operations of the parallelogram is thrice its height. If the area is 147 sq.units, so what is the value of its base and height?
4. A parallelogram has sides equal to 10m and 8m. If the distance between the shortest sides is 5m, and then find the distance between the longest sides of the parallelogram. (Hint: Start find the area of parallelogram using distance between shortest sides)