Triangle identities · trigonometric identity
½ a b sin C = ½ b c sin A — Triangle area in two pairings
When to use it
Any two-side-plus-included-angle area computation in a triangle. Written equal like this, it is the sine rule in disguise: the area does not care which pair of sides you use.
Why it is true
½·(one side)·(height to that side) computes the area whichever side you call the base; expressing each height with a sine gives three formulas for the same area, so they are equal.
The full line-by-line version is on the proof page for triangle area in two pairings; the “how would I find this myself” version is in the derivation.
Where it comes from
Divide to get the sine rule. This identity is how the sine rule falls out: cancel the common factor in the area equations.
- ½ a b sin C = ½ b c sin AEqual areas. · Triangle area in two pairings
- divide by ½ bCommon factor.
- a sin C = c sin A → a/sin A = c/sin CRearrange — the sine rule for two of the three pairs.
Worked examples
- A triangle has a = 7, b = 5 and included angle C = 60°. Find the area and then angle A via the sine ruleArea = 35√3/4; and the same area written with another base gives a sin C = c sin A
Checked at these angles
Two sides of the identity evaluated with the same engine that runs the verifier. A single line disagreeing would break the build.
| Angle | Left side | Right side | Agree |
|---|---|---|---|
| 0° | 0 | 0 | yes |
| 7.5° | 0.835475 | 0.835475 | yes |
| 15° | 1.732051 | 1.732051 | yes |
| 18° | 2.094148 | 2.094148 | yes |
| 22.5° | 2.628626 | 2.628626 | yes |
| 30° | 3.464102 | 3.464102 | yes |
| 37° | 4.138418 | 4.138418 | yes |
| 45° | 4.732051 | 4.732051 | yes |
| 53° | 5.093254 | 5.093254 | yes |
| 60° | 5.196152 | 5.196152 | yes |
| 67.5° | 5.078116 | 5.078116 | yes |
| 75° | 4.732051 | 4.732051 | yes |
| 90° | 3.464102 | 3.464102 | yes |
The letters are not free
a, b, c and A, B, C here are one triangle's sides and angles: A + B + C = 180° and the sides follow the sine rule. Feed the same six letters with random values and the equation fails — that is a property of the formula, not a bug in the tool.
The mistake students make
Treating a, b, c, A, B, C as six free numbers. They are not independent: A + B + C = 180° and the sides must follow the sine rule. Random values make this identity fail — that is exactly why our sampler generates real triangles.
Try it
- Verify this identity — both sides are pre-filled; change one character and watch the counterexample appear.
- Ask “which identity should I use?” about the left-hand side — this identity is what the chooser should rank first.
- Cheat sheet entry for triangle identities.
Related identities
- sin²θ + cos²θ = 1Pythagorean identitydetailsproof
- sin(A + B) = sin A cos B + cos A sin BSine of a sumdetailsproof
Category hub: Triangle identities · all identities: /identities