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Maths / Areas Of Parallelograms And Triangles / Area of Parallelograms
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Area of Parallelograms

Area of Parallelograms:  Area of a parallelogram is caluculated as the product of base and the corresponding height.

In the figure ABCD is a parallelogram where AL perp CD and CM perpAD. To calculate area of this parallelogrami if we consider the base as CD then the corresponding height will be AL because AL perp CD, but if we take base as AD then the corresponding height will be CM because CM perp AD. Thus

Area of parallelogram ABCD = Base X height

                                                      = AD X CM

                                                      = CD X AL

 

Theorem 1      The area of a parallelogram is the product of its base and the corresponding altitude.

Given      A parallelogram ABCD in which AB is the base and AL the corresponding altitude.

To prove     ar(parallel ^{gm}ABCD) = ABtimes AL

Construction     Draw BM perp produced DC to M. Now ALMB  is a rectangle.

Proof   Since (arparallel ^{gm}ABCD)  and rectangle ALMB   are on the same base and between the same parallels.

therefore               ar(parallel ^{gm}ABCD)     = ar(rect.ALMB)

                       = ABtimes AL                   [  The area of Rectangle = Base X Height ]

Hence,         (arparallel ^{gm}ABCD)=ABtimes AL

Illustration: I n the given figure, ABCD is a parallelogram, AL ⊥ DC and CM ⊥ AD. If AB = 14 cm. AL = 10 cm and CM = 7 cm, find AD.

Area of paralleogram ABCD = DC × AL  (Taking base as DC and Height as AL )

Area of paralleogram ABCD = AB × AL (AB = DC as opposite side of the parallelogram are equal)

Therefore,

Area of paralleogram ABCD = 14 × 10 ……(1)

Taking the base of Parallelogram ABCD as AD the height will be CM

Area of paralleogram ABCD = AD × CM (taking base as AD and height as CM)

Area of paralleogram ABCD = AD × 7 ……(2)

Since equation 1 and 2 both represent the Area of the same Parallelogram ABCD , both should be equal.

Hence from equation (1) and (2),

This means that,

  14 X 10 = AD X 7

  AD X 7 = 14 X 10

         AD=frac{14 X 10}{7}=frac{140}{7}= 20; cm


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