Interactive resource
Law of Sines: Circumcircle Proof a = 2R sin A
Geometry. Circumcircle, chord midpoint, and right triangle OMB — see sin A = (a/2)/R. Toggle to verify the cosine law too. Free interactive GeoGebra…
Overview
The geometric heart of the law of sines is the circumcircle: the central angle is twice the inscribed angle, the perpendicular from O bisects the chord, hence a = 2R sin A.
Explanation
Circumcenter O, radius R; M midpoint of side a = BC ⇒ OM ⊥ BC and BM = a/2.
Central angle ∠BOC = 2∠A, so in right triangle OMB: sin A = (a/2)/R, hence a = 2R sin A.
Thus a/sin A = b/sin B = c/sin C = 2R.
Law of cosines: c² = a² + b² − 2ab cos C (and cyclic).
How to use
- Open the scene (sine mode by default).
- Watch the circumcircle, center O, midpoint M, and green right triangle OMB.
- Drag A/B/C and compare (a/2)/R with sin A in the panel.
- Switch to cosine mode to check the residual near 0.
When to use which
SAS → cosines; ASA/AAS → sines.
Knowledge points
FAQ
What about right triangles?
With a right angle, the cosine law reduces to Pythagoras — compare with the Pythagorean tool.
Is this tool free?
Yes. It runs in the browser with no install.
Good for class demos?
Yes — project it and ask what changes when you drag a parameter.
How do I write the formula afterward?
Open the equation editor or symbol sheet and copy a clean expression.