Sine Triangle Calculator

Apply the Law of Sines to solve a triangle: enter two angles and the length of the side opposite one of them to find the missing angle, the other two sides, and the triangle's area.

Quick Facts

Law of Sines
a/sin(A) = b/sin(B) = c/sin(C)
Each side divided by the sine of the angle opposite it is the same constant.
Angle sum
A + B + C = 180°
Once two angles are known, the third follows immediately.
Area formula
Area = ½·a·b·sin(C)
Two sides and their included angle give the area with no height needed.

Your Results

Calculated
Angle C
-
C = 180° − A − B
Side b
-
b = a·sin(B)/sin(A)
Side c
-
c = a·sin(C)/sin(A)
Area
-
Area = ½·a·b·sin(C)

Ready

Enter two angles and the side opposite one of them, then press Calculate.

Formula and method for the Sine Triangle Calculator

The Law of Sines relates every side of a triangle to the sine of the angle opposite it: a/sin(A) = b/sin(B) = c/sin(C). This calculator uses the AAS/ASA form: you supply two angles (A and B) and one side (a, the side opposite angle A), and it solves for the rest of the triangle from that single relationship.

How the calculation works

Since the interior angles of any triangle sum to 180°, the third angle follows immediately: C = 180° − A − B. Because side a and angle A form a known ratio (a/sin A), that same ratio applies to the other two sides: b = a·sin(B)/sin(A) and c = a·sin(C)/sin(A). Once two sides and the angle between them are known, the area is found with the trigonometric area formula Area = ½·a·b·sin(C), which needs only two sides and their included angle — no separate height measurement required. Enter angles in degrees and keep the side length in one consistent unit; the calculated sides and area will use that same unit (and its square, for area).

Common mistakes

  • Mismatched side/angle pair: side a must be the side directly opposite angle A, not an adjacent side — the Law of Sines only pairs a side with the angle across from it.
  • Angles that don't form a triangle: A and B must each be positive and their sum must be less than 180°, otherwise no valid triangle exists.
  • Degrees vs. radians: this calculator expects angles in degrees; entering a radian value (like 1.05 instead of 60) will give a nonsensical result.

Real-world applications

  • Surveying and navigation use the Law of Sines to find distances that cannot be measured directly, from two observed angles and one known baseline.
  • Engineering and construction use it to solve triangular trusses and supports when only some sides and angles are measurable on site.
  • Astronomy and geodesy apply the same relationship (triangulation) to compute distances to distant or inaccessible points.
  • Trigonometry students use it to verify hand calculations and to build intuition for AAS, ASA, and SSA triangle problems.

Frequently Asked Questions

What is the Law of Sines?
The Law of Sines states that in any triangle, the ratio of a side to the sine of the angle opposite it is the same for all three sides: a/sin(A) = b/sin(B) = c/sin(C). It lets you solve a triangle when you know two angles and one side (AAS or ASA), or two sides and a non-included angle (SSA).
How do I find the third angle of a triangle?
The interior angles of any triangle always sum to 180°, so once you know two angles A and B, the third is C = 180° − A − B.
Why does the calculator need a side and its opposite angle?
The Law of Sines works with matched pairs of a side and the angle directly across from it. That known ratio (side ÷ sin of its opposite angle) is the constant used to solve for the other two sides, so you must enter one full pair, not just any side.
How is the triangle's area calculated?
Once two sides and the included angle are known, the area is Area = ½·a·b·sin(C), where C is the angle between sides a and b. This needs no separate height measurement.

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