Triangle identities · proof ·what it is and when to use it

Proof: ½ a b sin C = ½ b c sin A

Triangle area in two pairings — proved by same area, three choices of base. Every line below says which rule it uses, so nothing has to be taken on faith.

The proof

Take the area formula ½·base·height and write each height as a sine of the adjacent angle times the other side.

1. Area = ½ · b · h_b
   Base b, height to it.
2. h_b = a sin C
   The height from A onto side b, from the right triangle at C.
3. Area = ½ a b sin C
   Substitute.
4. Area = ½ b c sin A = ½ a c sin B
   Same argument with the other two sides as bases.
5. ½ a b sin C = ½ b c sin A
   Both equal the same area.

Where the proof stops applying

The same claim, checked numerically

A proof is not the same thing as a check, and this page shows both: the reasoning above, and the first few angles fed to the same engine behind the verifier. If they ever disagreed, the data would be broken — the build would fail before publishing (accuracy policy).

Related

• How would I find ½ a b sin C = ½ b c sin A myself? — the derivation, which is a different question from the proof.
• Triangle area in two pairings: when to use it — the practical side.
• All triangle identities · proof index

sampled at 800 random anglessampled over real triangles only (A+B+C = 180°)Source: Standard triangle geometry (OpenStax Precalculus, Ch. 7.4 Law of Sines); constraint checked numerically 2026-09-26 · revised 2026-09-26 ·how we check ·accuracy policy ·report an error