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(5-2sqrt(5)) and sqrt(5-sqrt(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)
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 2004-10-25 16:58:53