What is the diameter of this circle.?

Question

I could not describe this without a picture, so follow this link to answer the question. The one restriction is that you must use trig to solve this problem!!!

http://brianjordan.googlepages.com/areaofcircle

Answer 1

Bear with me I've solved it, but need to describe it to you carefully, so I'll do an edit to finish off:

Draw a line vertically to the 0.75 baseline. It meets the baseline at right-angles, but will not bisect it!

Label the left-half of the baseline with x and the right-hand bit with y.
Note that x + y = 0.75

Now draw a line joining the point of contact of the left-hand tangent and the centre of the circle, and another line joining the point of contact of the right-hand tangent and the cente of the circle.


You now have two kites; one on the left and one on the right.

Note that the 73° is the size of the angle of the left-hand kite at the centre of the circle, and the 77° angle is the same as the angle at the centre of the circle for the right-hand kite. Reason ? The angles at opposits ends of the kite add up to 180°, as do the 73° and the angle next to it on the baseline.


Now draw in the axes of symmetry of the kites - they bisect the opposite corner angles.

Let the radius be r, then in the lower left centre ∆,
r tan (73/2)° = x

Likewise, in the right-hand one,
r tan (77/2) = y

Add:
r tan (73/2)° + r tan (77/2) = x + y
r { tan (73/2)° + tan (77/2) } = 0.75
2r { tan (73/2)° + tan (77/2) } = 1.5

So diameter = 2r = 1.5 / { tan (73/2)° + tan (77/2) }

⇒ diameter ≃ 0.976946 units

Answer 2

the final packing is so as that the centres of the circles sort equilateral triangles. To p.c.. them like this, and facilitate counting on a similar time, place a million" circles so they sort 3 flippantly spaced diagonals, and sort 6 segments. This takes 6 * 4 + a million = 25 circles. Now fill interior the 6 segments with 6 each - it somewhat is all which will in good shape. So, 25 + 36 = sixty one. 6 * 10 + a million ??? Does that propose yet another fill/count selection technique? a standard distinctive for the diameter of the great circle will supply a one-in-the-centre filling development. a good distinctive will supply a various filling development.

Answer 3

This is straightfoward. We know that:

0.750 = r (Tan(77/2)+Tan(73/2)), so that the diameter is

(1.5)/(Tan(77/2)+Tan(73/2)) = 2r = D = 0.976946