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September 20, 2014

September 20, 2014

Posted by **Anonymous** on Sunday, May 20, 2012 at 9:54pm.

- Geometry -
**Reiny**, Monday, May 21, 2012 at 7:47amDraw 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.

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