Solve for d small diameter A=(\pi *D^(2) )/(4) - (\pi *d^(2) )/(4)

Question

Solve for d small diameter

`A=(\pi *D^(2) )/(4) - (\pi *d^(2) )/(4)`

Answer 1

`\boxed{\text{\Large √(-1.27324A+D^2)$}}`

Explanation:

`\displaystyle A=(\pi * D^2)/(4) - (\pi * d^2)/(4)`

`\displaystyle A=0.785398D^2 - 0.785398d^2`

Solve for d

`\displaystyle A-0.785398D^2= - 0.785398d^2`

`\displaystyle (A-0.785398D^2)/(- 0.785398) = d^2`

`-1.27324A+D^2=d^2`

`\displaystyle √(-1.27324A+D^2) =d`

Answer 2

`\boxed{ \tt{d = \sqrt{D^(2) -(4A)/(\pi)} }}`

Explanation:

`if \: A=(\pi *D^(2) )/(4) - (\pi *d^(2) )/(4) \\ then \to \\ A = (\pi)/(4) (D^(2) - {d}^(2) ) \\ (D^(2) - {d}^(2) ) \pi = 4A \\ D^(2) - {d}^(2) = (4A)/(\pi) \\ {d}^(2) = D^(2) -(4A)/(\pi) \\ d = \sqrt{D^(2) -(4A)/(\pi)}`