04 // Interactive Formula Compendium // GEOMETRY-TRIGa / sin(A) = b / sin(B) = c / sin(C) = 2R
Law of Sines Formula — AAS, ASA & Circumradius Reference Sheet
The Law of Sines states that in any triangle with sides a, b, c opposite angles A, B, C, the ratio a/sin(A) = b/sin(B) = c/sin(C) = 2R is invariant, where R is the radius of the circumscribed circle.
Formula Reference
Main Formula
a / sin(A) = b / sin(B) = c / sin(C) = 2R
Rearranged Forms
b = a · sin(B) / sin(A) (Side Solver)
sin(B) = b · sin(A) / a (Angle Solver)
Area = ½ ab · sin(C) = abc / (4R)
Variables & Units
| Symbol | Name | Unit |
|---|---|---|
| b | Opposite Side Length (b) | m |
| a | Known Side Length (a) | m |
| A_deg | Opposite Angle A (degrees) | deg |
| B_deg | Target Angle B (degrees) | deg |
Inputs & Parameters
[m]
[deg]
[deg]
Calculated Result (Opposite Side Length (b))
b =16.91958m
Formula Notes & Background
- Drop an altitude h_c from vertex C perpendicular to side c.
- From the two resulting right triangles, h_c = b sin(A) and h_c = a sin(B).
- Equating b sin(A) = a sin(B) and dividing by sin(A)sin(B) yields a/sin(A) = b/sin(B); repeating from vertex A extends the equality to c/sin(C) = 2R.
Step-by-Step Examples
#1: Solve for side b when a = 12 m, A = 40°, and B = 65° (AAS triangle).
a = 12 mA = 40°B = 65°
b = 12 × sin(65°) / sin(40°) = 12 × 0.90631 / 0.64279
→ b = 16.919 m (and circumdiameter 2R = 18.669 m)
#2: Find the circumradius R of an equilateral triangle with side a = 10 m and A = 60°.
a = 10 mA = 60°
2R = 10 / sin(60°) = 10 / (√3/2) = 20 / √3
→ R = 10 / √3 ≈ 5.7735 m
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Frequently Asked Questions
What is the SSA ambiguous case in the Law of Sines?+
Because sin(θ) = sin(180° − θ), solving sin(B) = b sin(A)/a when given two sides and a non-included angle (SSA) can yield zero, one, or two valid triangles (B and 180° − B).
How is the Law of Sines connected to the circumscribed circle of a triangle?+
By the inscribed angle theorem, a chord of length a subtending inscribed angle A satisfies a = 2R sin(A), so the common ratio a/sin(A) equals the circumdiameter 2R.
How do you compute triangle area from SAS using sine?+
Since altitude h = b sin(C), the area of any triangle is Area = ½ a b sin(C).