Finding the area of a circle...with a twist!

Question

anonymous · Answer

attachment

anonymous · Answer

$\angle AOC = 90^\circ $

anonymous · Answer

The angle AOC is twice the angle ABC.
Now use Pythagoras theorem.

accessdenied · Answer

Cool problem.  :)

  Basically, we want to find the radius.  If we look at it a bit, we can see two radii as the sides of triangle AOC, so that triangle must be isosceles.
  If we use the "Central Angle - Inscribed Angle" idea that the central angle is twice the inscribed angle, we can find the angle that triangle AOC opens at the vertex.
  Then, you have to use a bit of trigonometry to find the radius from that isosceles triangle, and then area formula should work out from there.

anonymous · Answer

You don't need trig, because the central angle is a special angle :)

anonymous · Answer

$2r^2=9 \implies r = 1.22474 $

Area $= 14.1372 \; ft^2 $

anonymous · Answer

Awesome. thanks guys!

anonymous · Answer

:)

accessdenied · Answer

You're welcome!
I usually just refer to general triangle magics as Trigonometry.  :P