Area of a Parallelogram

A parallelogram is a quadrilateral whose opposite sides are equal length and parallel. The diagonals of a parallelogram are unequal and bisect each other. If ABCD is a parallelogram then its area can be calculated in two ways:

(a) When the base and height are given:

Since, ABCD is a parallelogram with base, AB = b and height, h = DL


parallelogram-01

\therefore Area of the parallelogram ABCD  = Area of the rectangle DLMC
Area of the parallelogram  = LM \times DL
Area of the parallelogram  = AB \times h (as LM = AB)
Area of the parallelogram  = b \times h
\therefore Area of parallelogram  = base  \times height

(b) When adjacent sides and their included angle is given:

Since, ABCD is a parallelogram. Take AB and CD as its two adjacent sides with included angle \theta .


parallelogram-02

\therefore Area of parallelogram  = AB \times DL = b \times h
But \frac{h}{{AD}} = \sin \theta
Or h = AD\sin \theta
Or h = c\sin \theta
\therefore Area of ABCD = h \times c\sin  \theta
\therefore Area of Parallelogram  = Product of adjacent sides \times \sin \theta

Example:

Find the base of a parallelogram whose area is 256square cm and height 32cm.

Solution:
Since, area  = 256square cm
height  = 32cm
\therefore Area of parallelogram  = base  \times height
\therefore 256 = base  \times 32
Base  = \frac{{256}}{{32}} = 8cm.

Example:

Find the area of a parallelogram, two adjacent sides of which are 17cm and 20cm and their included angle is{60^ \circ }.

Solution:
Here, one side, b = 17cm, other side, c = 20cm, \theta  = {60^ \circ }
\therefore Area  = bc\sin \theta
 = 17 \times 20 \times  \sin {60^ \circ }
 = 340 \times 0.866
 = 294.44square cm

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