Calculate radius if two chords and angle between them are given

**Garas** · June 22nd 2010, 02:56 PM

Two chords AB=4 and AC=7 are given in circle. The angle between them is 60. Calculate radius.

**undefined** · June 22nd 2010, 03:31 PM

Originally Posted by Garas

Two chords AB=4 and AC=7 are given in circle. The angle between them is 60. Calculate radius.

We can use law of cosines to get the third side, and then the equation for diameter of circumcircle on this page.

$\displaystyle \text{diameter} = \frac{abc}{2\cdot\text{area}} = \frac{2abc}{\sqrt{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}}$

**earboth** · June 22nd 2010, 10:22 PM

Originally Posted by Garas

Two chords AB=4 and AC=7 are given in circle. The angle between them is 60. Calculate radius.

Here comes a slightly different approach:

1. Calculate the 3rd side as described by undefined.

2. Use the theorem of the circumscribed circels of a triangle: The central angle is twice as large as the angle at the circle. Use the right triangle at the center of the circle and half of the 3rd side to calculate the length of the radius:

3. $r=\dfrac{\frac12 BC}{\sin(60^\circ)}$

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