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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 $\perp$AD. 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. ThusArea 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     $ar(\parallel ^{gm}ABCD) = AB\times AL$Construction     Draw BM $\perp$ produced DC to M. Now $ALMB$  is a rectangle.Proof   Since $(ar\parallel ^{gm}ABCD)$  and rectangle $ALMB$   are on the same base and between the same parallels.$\therefore$               $ar(\parallel ^{gm}ABCD)$     $= ar(rect.ALMB)$                       $= AB\times AL$                   [  The area of Rectangle = Base X Height ]Hence,         $(ar\parallel ^{gm}ABCD)=AB\times AL$

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 CMArea 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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