Theorem

Sine rule, with the circumradius

Trigonometry • Triangle

Formula

\frac{a}{s i n A} = \frac{b}{s i n B} = \frac{c}{s i n C} = 2 R

In any triangle

A B C

with circumradius

R

\frac{a}{s i n A} = \frac{b}{s i n B} = \frac{c}{s i n C} = 2 R

, so each side equals

2 R

times the sine of the angle opposite it.

How to use it

1Pair each side with the angle opposite it: side $a$ is $B C$ and goes with the angle at $A$ .
2Given one angle and the circumradius, the opposite side is $2 R sin A$ in one step.
3A side equal to $2 R$ forces its opposite angle to be $9 0^{\circ}$ , since a diameter subtends a right angle.

In a triangle the angle at

A

4 5^{\circ}

and the circumradius is

8

cm. How long is the side opposite

A

Step-by-step

1
$a = 2 R sin A = 2 \times 8 \times sin 4 5^{\circ}$
Why: Apply the sine rule in its circumradius form.
2
$= 16 \times \frac{1}{2} = 82$ cm

Answer

82

Watch out

The factor is

2 R

and not

R

— the commonest slip. The side used must be the one opposite the angle used; pairing a side with an angle next to it gives nothing.

Builds on

Also known as

law of sines, sine rule, a = 2R sin A, extended law of sines, circumradius and side