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Octagon in a Circle (Posted on 2004-10-25) Difficulty: 3 of 5
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.

No Solution Yet Submitted by Victor Zapana    
Rating: 3.5000 (2 votes)

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Exact solution | Comment 2 of 6 |

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

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