Hints and solutions to Problem of the day for Monday, May 11

A (regular) hexagon is inscribed in a circle, which is inscribed in a square of side 10 cm. What is the length of each side of the hexagon?

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• I guess you can tell that this is the week for picture problems!
  

• Since the sides of the square are 10 cm long, the radius of the circle is 5 cm. If you draw two radii from the center of the circle to adjacent corners of the hexagon as follows:
  

  then there is a triangle formed, with two black sides (the radii), and one green side (a side of the hexagon).

• The two black sides of the triangle, being radii of the circle, are each 5 cm long. And the angle between them is 60 degrees, because there are six such angles around the center of the circle, and 1/6 of 360 degrees is 60 degrees.
• So our triangle is isosceles and has a 60 degree angle between its two equal sides.
• But the angles opposite the equal sides of an isosceles triangle must also be equal - and the three angles of the triangle add up to 180 degrees. That means these two angles must also be 60 degrees. So our triangle is not just isosceles, it is equilateral. So all three of its sides are 5 cm long.
• But the green side of the triangle is a side of the hexagon - so each side of the hexagon is 5 cm long.