A parallelogram containing a `60^0` angle has perimeter `p` and its longer diagonal is of length `ddot` Find its area.

Explanation

Perimeter,
`2(x+y)=p`
or `x+y=p/2`
Also, `d^2=(x+ycostheta)^2+y^2sin^2theta …(i)`
`=x^2+y^2+2xycostheta`
`=x^2+y^2+xy (theta=60^@)`
`=(x+y)^2-xy`
`=p^2/4-xy` [using (i)]
or `xy=(p^2)/4-d^2`
Now,
Area of parallelogram `=xysin60^@`
`=(p^2/4-d^2)sqrt3/2=sqrt3/8(p^2-4d^2)`