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POW Solution, Jan. 1014
Posted:
Jan 17, 1994 10:26 AM


****************************** Problem of the Week, January 1014
A regular HEXAGON and an equilateral triangle are both inscribed in the same circle so that the hexagon and the triangle share three vertices. The RADIUS of the circle is 8 units. What is the area of the region between the two polygons?
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We received only one solution to the problem of the week this time around! The winner is:
Dan Hirschhorn, Illinois State University
Please note the first line of Dan's submission!
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From: danh@spider.math.ilstu.edu (Dan Hirschhorn)
I assume you'll get lots of good stuff from the studentshere is my own take:
If you draw in the radii to the vertices of the equilateral triangle, you get 6 triangles all congruent by SAS (8120 deg8). 3 make up the equilateral triangle, 3 are outside of it, so the area of region desired is 3 * area of 1 of these 8120 deg8 triangles or 1/2 (area of hexagon) or the area of the equilateral triangle. I choose 1/2 hexagon to make it interesting. Draw all the radii to the vertices of the hexagon. You form 6 equilateral triangles each with side of 8. 3 of these triangles equal 1/2 the hexagon. Using formula "side squared root 3 over 4" for the area of an equilateral triangle , you get 48 sqt (3) as desired area.



