
Evan H. answered 11/11/14
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The diameter of a full circle is 2pi*r, for half a circle it is pi * r. We can see that the arc length covered going around a circle is simply theta * r.
And the area is theta/2 * r^2
Now if you plug in, you get
2.4 = theta * r + 2 r = r(theta+2)
0.25 = (theta/2) * r^2
so theta = 2.4/r -2
theta/2 = 1.2/r - 1
Plug this into the area equation to get:
.25 = (1.2/r - 1) * r^2
.25 = 1.2 r - r^2
-r^2 + 1.2 r = .25
r^2 - 1.2r = -.25
r^2 - 6r + .25 = 0
Quadratic equation:
1.2 +- sqrt(1.44 - 1)/ 2
(1.2 +- .66)/2
r = .93 or r = .27
Now we can only have one, so we need to see if one of them gives us a central angle of more than pi/2(90 degrees)
Going back to the first equation:
2.4 = r(theta + 2)
2.4/r - 2 = theta
2.4/(0.27) - 2 = theta or 2.4/(.93) - 2 = theta
The first one gives ~2pi radians which is much too large, the second one gives ~.58 radians or 33ish degrees which works.
Disclaimer: The exact numbers may be wrong, but the process is correct. I did this without pencil and paper.