Circle O is inscribed is square ABCD, and at the same time, is circumscribed about square PQRS. WHich are is the smaller area, the region inside the inscribed circle minus the area of square PQRS or the are inside square ABCD minus the area of the inscribed circle? The length of the side of square ABCD measures two feet.

Answer 1

Draw you diagram so that the vertices of square ABCD and square PQRS are in corresponding order, that is AB is || to PQ
Let M be the midpoint of AB and N be the midpoint of PQ, and let O be the centre of the circle
Joine AC, which will pass through PR
AM =1, and OM = 1

area of ABCD = 4, area of circle = π(1^2) = π
area of region between outer square and circle = 4-π
= appr .8584

PO = 1, let ON = x
x^2 + x^2 = 1^2
2x^2 = 1
x^2 = 1/2
x = 1/√2
so ON = 1/√2 ---> PN = 1/√2 ---> PQ = 2/√2
area of inside square = (2/√2)^2= 2
difference between circle and inside square = π - 2
= appr 1.1416

BTW, this is the approach Archimedes took in finding an approximation to π .
He took successive differences in areas between
incribed and circumsribed squares, octogons, 16-sided polygons, 32-sided polygons, etc.