Law of Sines Calculator
The missing side or angle from the sine rule, with the common ratio and the circumradius, and a warning when two triangles fit.
Results
The one that was missing
14.1421
The common ratio, a ÷ sin A
20.0000
Circumradius, the circle through the corners
10.0000
m
Diameter of that circle
20.0000
m
The other angle that also fits
—
°
How many answers fit
Only one
What it is · how much
a ÷ sin A 20
2R 20
R 10
What this tool does
The sine rule says a side divided by the sine of its opposite angle is the same for all three pairs — and that common value is the diameter of the circle through all three corners. Finding a side from it is always safe. Finding an angle is not: a sine has two possible angles under 180°, so there can be two triangles, and the page tells you when that happens.
Formula
a ÷ sin A = b ÷ sin B = c ÷ sin C = 2R
Variables
| Symbol | Meaning | Unit |
|---|---|---|
a | A side you know | m |
alpha | The angle opposite that side | ° |
find | What you are looking for | — |
beta | The angle opposite the side you want | ° |
b | The side opposite the angle you want | m |
OU | The one that was missing | — |
RA | The common ratio, a ÷ sin A | — |
CR | Circumradius, the circle through the corners | m |
DI | Diameter of that circle | m |
SD | The other angle that also fits | ° |
NS | How many answers fit | — |
TB | What it is · how much | — |
Worked example
- A side you know10 m
- The angle opposite that side30 °
- What you are looking forside
- The angle opposite the side you want45 °
- The side opposite the angle you want8 m
- The one that was missing14.1421
- The common ratio, a ÷ sin A20.0000
- Circumradius, the circle through the corners10.0000 m
- Diameter of that circle20.0000 m
- How many answers fitOnly one
- What it is · how mucha ÷ sin A 20 2R 20 R 10
Limitations
- The calculation runs at full precision and only the display is rounded. If you copy an intermediate value and retype it, small differences can appear.