Last updated: July 31, 2026
Law of Sines Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
C = 180°-A-B, k = a/sin(A), b = k sin(B), c = k sin(C), Area = 1/2 ab sin(C)
Where:
- A,B,C=Triangle angles(°)
- a,b,c=Opposite side lengths
- k=Law of sines scale factor
Worked Examples
30°-60°-90° triangle from side a
Use A = 30°, B = 60°, and a = 5.
- 1C = 180 - 30 - 60 = 90°.
- 2k = 5/sin(30°) = 10.
- 3b = 10 sin(60°) ≈ 8.660254 and c = 10 sin(90°) = 10.
- 4Area = 1/2 × 5 × 8.660254 × 1 ≈ 21.650635.
Balanced acute triangle
Use A = 45°, B = 45°, and a = 8.
- 1C = 90°.
- 2Equal angles A and B give equal sides a and b.
- 3c is the hypotenuse-like longest side in this case.
Invalid angle sum
The first two angles must sum to less than 180°.
- 1Compute A + B first.
- 2100 + 90 = 190° exceeds the triangle limit.
- 3The calculator returns an error.
Introduction
This law of sines calculator solves the AAS triangle case from two angles and one opposite side. It first finds the third angle, then uses the common sine ratio to recover the remaining sides and the triangle area.
AAS Triangle Setup
This calculator is designed for the angle-angle-side case.
You provide two angles A and B.
You also provide side a opposite angle A.
The third angle C follows from the triangle sum.
Then the remaining sides come from the law of sines.
The Shared Sine Ratio
The law of sines creates one common scale factor for all sides.
k = a/sin(A).
Once k is known, every other side is k times the sine of its opposite angle.
This keeps the computation compact and easy to verify.
The calculator reports side b and side c directly.
Finding the Third Angle
The triangle angle sum is always 180°.
C = 180° - A - B.
If A + B is not less than 180°, no triangle exists.
A positive C is required for a valid result.
Angle C is returned in degrees.
Area Formula
After solving side b, the area follows from a standard trigonometric formula.
Area = 1/2 ab sin(C).
This works because sides a and b meet at angle C.
For the sample case C = 90°, the sine factor becomes 1.
The area is shown alongside the solved dimensions.
Worked Example
The sample A = 30°, B = 60°, a = 5 gives a clean 30-60-90 triangle.
C = 90°.
k = 10.
b ≈ 8.660254 and c = 10.
Area ≈ 21.650635.
Where It Helps
AAS data appears often in navigation, surveying, and classroom trigonometry.
Indirect measurement problems.
Sketch-based geometry exercises.
Triangle reconstruction from partial data.
Manual verification of solved triangles.
FAQs
What triangle case does this solve?
It solves the AAS case: two angles and one non-included side opposite angle A.
Why must A + B be less than 180°?
Because the third angle C must remain positive in any valid triangle.
How is side c found?
After finding C and the scale factor k = a/sin(A), compute c = k sin(C).
Does the calculator also return side b?
Yes. It returns side b, side c, angle C, and area.
Can I use decimal angles and sides?
Yes. Decimal values are fully supported.
What happens if sin(A) would be zero?
That corresponds to an invalid angle input because A must be strictly between 0° and 180°.