The octagon ABCDEFGH is inscribed in a circle, with the vertices around the circumference in the given order. Given that the polygon ACEG is a square of area 5 and the polygon BDFH is a rectangle of area 4, find the maximum possible area of the octagon.
Area = 3sqrt(5)
From the square, the circle radius = sqrt(5/2)
Call the sides of the rectangle x and 4/x. Create the right triangle from the center of the circle to the corner of the rectangle.
By pythagorean theorem (x/2)^2 + (2/x)^2 = 5/2
Solutions sqrt(8) and sqrt(2) These are the sides of the rectangle.
The ocatgon of greatest area is the one which most closely approximates a circle. This means none of its sides should be too large. The smallest they can be is when the sides of the rectangle are at a 45 degree angle to the sides of the square.
I placed this figure on a coordinate plane with the square's corners on the axes and the center at the origin. The rectangle is then orthogonal. Using pythagorean again the sides alternate in length between sqrt(52sqrt(5)) and sqrt(5sqrt(5))
Pythagorean theorem again yields apothems of sqrt(5+2sqrt(5))/2 and sqrt(5+sqrt(5))/2 respectively.
The areas of the central triangles are then sqrt(5)/4 and sqrt(5)/2 respectively.
The total for these 8 central triangles (4 of each) is 3sqrt(5)
Jer
p.s. An apothem is a perpendicular segment from the center of a circle to a chord.
Edited on October 25, 2004, 5:00 pm

Posted by Jer
on 20041025 16:58:53 