geo is hard

We can use the  secant-secant  product theorem  to find PC

AC (AC + PC)  = AB (AB + SB)

8 ( 8 + PC)  = 5 (5 + 25)

64 + 8PC  =  5 (30)

8PC =  150 - 64

8PC = 86

PC  =  86/8  =  43/4  =10.75

So   AP  = 8 + 10.75  = 18.75

So  SP^2  =   [ AS^2 - AP^2 ] =   [ 30^2  - 18.75^2] 

Connect   SC  and we  have right triangle  SCP....and since APS is a right angle....then SC will be the diameter of the  circle

So  SC   = sqrt  [ PC^2  + SP^2 ]  =  sqrt  [ 10.75^2  + 30^2  - 18.75^2  ]   = sqrt (664)

So....the radius =  sqrt (664) / 2

And the area of the  circle is     (sqrt (664) / 2 )^2 * pi  =   (664/4)  pi = 166 pi  units^2

AC * AP = AB * AS

8(8 + PC) = 5 * 30

PC = 10.75

SP = sqrt(AS² - AP²)

SC = sqrt(SP² + PC²)

Area of a circle       A = (SC/2)² * pi