ASA Triangle Calculator
The ASA Triangle Calculator solves a triangle when you know two angles and the length of the side between them, the classic Angle-Side-Angle case used throughout trigonometry, surveying, engineering drawing and NCEA maths. You enter angle A and angle C in degrees, plus the length of side b, the included side that runs between those two angles, in any consistent unit such as centimetres or metres. The calculator first finds the third angle, B, by subtracting A and C from 180 degrees, then applies the Law of Sines to work out the two remaining sides, a and c. It returns angle B, the area in square units, the perimeter, and whether the triangle is acute, right or obtuse, plus a detailed breakdown listing every side and every angle. A worked example using the default values of 45 degrees, 10 units and 45 degrees walks through each step of the calculation, so you can check your own inputs against it. Results update instantly as you change any value, and a warning appears if angle A and angle C are not both greater than zero or their sum reaches 180 degrees or more, since no triangle can exist under those conditions. Because ASA always defines exactly one triangle, unlike the ambiguous SSA case, any valid combination you enter produces a single, unique answer.
1. Known Angles
Angles A and C must each be greater than 0° and their sum must be less than 180°.
2. Included Side
Side b connects vertex A to vertex C. It is opposite angle B (the unknown angle). Enter any consistent unit (cm, m, mm, etc.).
All Sides
All Angles
Angle B = 180 - 45 - 45 = 90°
Side a = 10 × sin(45°) / sin(90°) = 10 × 0.7071 / 1.0000 = 7.07 units
Side c = 10 × sin(45°) / sin(90°) = 7.07 units
Area = (10² × sin(45°) × sin(45°)) / (2 × sin(90°)) = (100 × 0.5) / 2 = 25.00 sq units
Perimeter = 7.07 + 10 + 7.07 = 24.14 units
How to Solve an ASA Triangle
The ASA (Angle-Side-Angle) case occurs when you know two angles of a triangle and the side that lies between them. Because the angles of any triangle sum to 180 degrees, knowing two angles immediately tells you the third. Once all three angles are known, the Law of Sines links every angle to its opposite side, so the full triangle can be solved from just one known side.
The Formulas
Label the triangle with angles A, B, C at the respective vertices, and sides a, b, c opposite to those vertices. In the ASA case you know angles A and C and the side b (which lies between them, opposite angle B).
| Step | Formula | Notes |
|---|---|---|
| Find angle B | B = 180 - A - C | Angles in a triangle sum to 180° |
| Find side a | a = b × sin(A) / sin(B) | Law of Sines |
| Find side c | c = b × sin(C) / sin(B) | Law of Sines |
| Area | Area = (b² × sin(A) × sin(C)) / (2 × sin(B)) | Derived from half base times height |
| Perimeter | P = a + b + c | Sum of all sides |
The Law of Sines
The Law of Sines states that in any triangle: a / sin(A) = b / sin(B) = c / sin(C). This ratio is constant for a given triangle and equals twice the circumradius (the radius of the circle that passes through all three vertices). Once you know one side and all three angles, you can rearrange this equality to find any unknown side.
Triangle Types by Angle
Knowing all three angles lets you classify the triangle:
- Acute triangle: all three angles are less than 90°.
- Right triangle: one angle is exactly 90°.
- Obtuse triangle: one angle is greater than 90°.
- Equilateral triangle: all three angles equal 60° (and all sides are equal).
- Isosceles triangle: two angles are equal (and their opposite sides are equal).
When ASA Cannot Be Solved
The ASA case always produces a valid, unique triangle provided: both given angles are greater than 0°, their sum is strictly less than 180°, and the included side is positive. If the two angles sum to 180° or more, no triangle is possible. Unlike the SSA (side-side-angle) case, ASA never produces an ambiguous solution.
Related Calculators
- Maths and Stats Calculators: full hub for maths tools.
- Triangle Calculator (general): solve from any combination of sides and angles.
- AAS Triangle Calculator: two angles and a non-included side.
- AAA Triangle Calculator: three angles known (similar triangles).
- SOHCAHTOA Solver: right-triangle trigonometry.
Method: Standard Euclidean geometry. Third angle derived from the angle sum property (A + B + C = 180°). Unknown sides calculated using the Law of Sines (a / sin(A) = b / sin(B)). Area formula: (b² × sin(A) × sin(C)) / (2 × sin(B)). All calculations use double-precision floating-point arithmetic.