Law of Sines: Missing Sides and the Circumradius

From one side and two angles (in degrees), finds the remaining sides and the circumradius with the law of sines a ÷ sin A = b ÷ sin B = 2R.

Explanation

The law of sines says that each side of a triangle, divided by the sine of the angle facing it, gives the same value, and that this shared value is the diameter $2R$ of the circle through all three vertices. One side plus two angles is enough to pin down everything else.

\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C} = 2R

$a$ — the side whose length you know
$A$ — the angle opposite $a$ , in degrees
$B$ — a second angle, in degrees; the third is $C = 180^\circ - A - B$
$R$ — the circumradius, the radius of the circle through the three vertices

Example

Take $a = 8$ , $A = 45^\circ$ and $B = 60^\circ$ , so $C = 180^\circ - 45^\circ - 60^\circ = 75^\circ$ . Start with the shared ratio:

2R = \dfrac{a}{\sin A} = \dfrac{8}{\sin 45^\circ} = 8\sqrt{2} \approx 11.3137

The remaining sides follow: $b = 2R \sin B \approx 11.3137 \times 0.8660 \approx 9.7980$ and $c = 2R \sin C \approx 11.3137 \times 0.9659 \approx 10.9282$ , with a circumradius of $R \approx 5.6569$ .

Notes

Angles are in degrees. Both $A$ and $B$ must be positive with $A + B < 180^\circ$ ; otherwise no triangle exists and the calculator says the angle is out of range.
Side $a$ and angle $A$ have to be an opposite pair. Pairing a side with the wrong angle changes the answer.
Two angles fix the shape of the triangle and the known side fixes its size, so the result is unique.
Starting instead from two sides and a non-included angle can leave two different triangles that both fit, the classic ambiguous case. Working from one side and two angles, as here, sidesteps it.