Law of Sines Calculator

Law of Sines is evaluated from Side a, Angle A and Side b. The calculation reports Side a, Side b and Side c.

Results

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About the Law of Sines Calculator

Law of Sines is treated here as a quantitative relation between Side a, Angle A, Side b and Angle B and Side a, Side b, Side c and Angle A.

The calculator uses a multi formula configuration. Each reported value is read as a direct evaluation of the stored rules with the declared field formats and units.

Formula basis:
a/sin(A) = b/sin(B) = c/sin(C)
b = a x sin(B)/sin(A)
C = 180 deg - A - B

Interpret the outputs in the order shown by the result fields. Optional inputs affect only the outputs that depend on those variables.

Formula & How It Works

The calculation applies the following relations exactly as recorded in the metadata:

a/sin(A) = b/sin(B) = c/sin(C)
b = a x sin(B)/sin(A)
C = 180 deg - A - B

Each output field is produced by substituting the supplied inputs into the relevant relation and then applying the declared rounding or text format.

Worked Examples

Example 1: AAS — Surveying Triangulation

Inputs

side_a: 450 angle_A: 52 angle_B: 73
Side a: 450. Side b: 546.10567. Side c: 467.783489. Angle A: 52 deg. Angle B: 73 deg. Angle C: 55 deg. Area: 100,652.3045 units^2

With Side a = 450, Angle A = 52 and Angle B = 73 as the stated inputs, the result is Side a = 450, Side b = 546.10567 and Side c = 467.783489. Each value corresponds to the declared output fields.

Example 2: ASA — Navigation — Coast Guard

Inputs

angle_A: 65 side_b: 12 angle_B: 75
Side a: 11.259346. Side b: 12. Side c: 7.985552. Angle A: 65 deg. Angle B: 75 deg. Angle C: 40 deg. Area: 43.4242 units^2

With Angle A = 65, Side b = 12 and Angle B = 75 as the stated inputs, the result is Side a = 11.259346, Side b = 12 and Side c = 7.985552. Each value corresponds to the declared output fields.

Example 3: SSA — Ambiguous Case

Inputs

side_a: 7 side_b: 10 angle_A: 35
Side a: 7. Side b: 10. Side c: 7. Angle A: 35 deg. Angle B: 0 deg. Angle C: 145 deg. Area: 20.0752 units^2

With Side a = 7, Angle A = 35 and Side b = 10 as the stated inputs, the result is Side a = 7, Side b = 10 and Side c = 7. Each value corresponds to the declared output fields.

Example 4: Area from Two Sides and Included Angle

Inputs

side_a: 8 side_b: 11 angle_C: 40
Side a: 8. Side b: 11. Side c: 0. Angle A: 0 deg. Angle B: 0 deg. Angle C: 40 deg. Area: 28.2827 units^2

With Side a = 8, Side b = 11 and Angle C = 40 as the stated inputs, the result is Side a = 8, Side b = 11 and Side c = 0. Each value corresponds to the declared output fields.

Common Use Cases

  • Solve triangle from two angles and one side
  • Find unknown side in an oblique triangle
  • Calculate navigation bearing and distance
  • Determine satellite positioning from angles