# the diagonal of a parallelogram are 18 cm and 30cm respectively.one side of a parallelogram is 12cm. find the area of a parallelogram?

Relevance
• Refer to the figure :

In any parallelogram, the two diagonals ( in our case AC and BD ) bisect each other

(say at O) such that areas of the four triangles are equal to each other.  Hence --

Area (A)  of whole of the parallelogram = 4 * (Area of any of the triangles)

=>  A  =  4 * ( Area of triangle AOB )  ........................ (1)

For the area of Triangle AOB we will use HERON's formula. According to this

formula -- If sum of all the three sides = P then ,

P/2 is known as Semi-perimeter and denoted as ' s '

Then area of the triangle --

= √[ s ( s - a ) ( s - b ) ( s - c ) ]

Where a, b and c are measurement of the three sides.

In our case  s  = (1/2) ( 9 + 12 + 15 ) = 18 cm

Hence Area of triangle AOB = √ [ 18 ( 18 - 9 ) ( 18 - 12 ) ( 18 - 15 ) ] cm²

=> √[ 18 * 9 * 6 * 3 ]  =  54 cm²

Hence Area of the Parallelogram  =  4 * 54  =   216 cm²  ............... Answer • The parallelogram is comprised of four triangles of equal area, and two of these triangles have sides 9, 12, 15 (a right triangle!), so the area of the parallelogram is 4×½×9×12 = 216 cm². • Refer to the figure below.

ABCD is a parallelogram with AB = 12 cm.

The diagonals AC (18 cm) and BD (30 cm) meet at M.

The two diagonals of parallelogram bisect each other.

Hence, AM = 18/2 = 9 cm

and BM = 30/2 = 15 cm

Use Heron's formula to calculate the area of ΔABM:

s = (12 + 15 + 9)/2 = 18

Area of ΔABM = √[18 × (18 - 12) × (18 - 15) × (18 - 9)] = 54

The two diagonals divide the parallelogram into 4 triangles with equal areas.

Area of the parallelogram = 54 × 4 = 216 (square units) 