Two sides and included angle (SAS) — in any length and angle units.
a² = b² + c² − 2bc·cos(A) | a/sin(A) = b/sin(B) = c/sin(C) | Area = √[s(s−a)(s−b)(s−c)]
The calculator applies the appropriate formula based on what you know:
Law of Cosines (for SSS or SAS):
a² = b² + c² − 2bc·cos(A)
Law of Sines (for ASA, AAS, or SSA):
a / sin(A) = b / sin(B) = c / sin(C)
Angle Sum Property:
A + B + C = 180°
Area via Heron's Formula (for SSS):
s = (a + b + c) / 2
Area = √[s(s−a)(s−b)(s−c)]
Area via SAS:
Area = (1/2) · b · c · sin(A)
Perimeter:
P = a + b + c
Step 1 — Find side a using the Law of Cosines: a² = b² + c² − 2bc·cos(A) a² = 7² + 10² − 2(7)(10)·cos(60°) a² = 49 + 100 − 140 · 0.5 a² = 149 − 70 = 79 a = √79 ≈ 8.888 Step 2 — Find angle B using the Law of Sines: sin(B)/b = sin(A)/a sin(B) = 7 · sin(60°) / 8.888 sin(B) = 7 · 0.8660 / 8.888 ≈ 0.6817 B = arcsin(0.6817) ≈ 42.95° Step 3 — Find angle C: C = 180° − 60° − 42.95° = 77.05° Step 4 — Area via SAS: Area = (1/2) · b · c · sin(A) = 0.5 · 7 · 10 · sin(60°) = 35 · 0.8660 ≈ 30.31 square units Step 5 — Perimeter: P = a + b + c = 8.888 + 7 + 10 = 25.888 ≈ 25.89 units
Result: Side a ≈ 8.89, Angle B ≈ 42.95°, Angle C ≈ 77.05°, Area ≈ 30.31 sq units, Perimeter ≈ 25.89 units
The calculator returns all three side lengths, all three angles, the area, and the perimeter of the triangle. Results are shown rounded to two decimal places by default. If you entered an SSA case, the calculator will warn you if an ambiguous (two-solution) triangle exists and display both valid solutions.
A triangle has six elements: three sides (a, b, c) and three angles (A, B, C). Knowing at least three of these elements (with at least one being a side) is sufficient to determine the entire triangle, provided they satisfy the triangle inequality.
| Case | Known | Method | |------|-------|--------| | SSS | All 3 sides | Law of Cosines, then Law of Sines | | SAS | 2 sides + included angle | Law of Cosines | | ASA | 2 angles + included side | Angle sum, then Law of Sines | | AAS | 2 angles + non-included side | Angle sum, then Law of Sines | | SSA | 2 sides + non-included angle | Law of Sines (may be ambiguous) |
The SSA (ambiguous) case can produce zero, one, or two valid triangles depending on the values given. The calculator handles this automatically and notifies you of multiple solutions.
Generalizes the Pythagorean theorem to any triangle. When angle A = 90°, it reduces to the familiar a² = b² + c².
Allows you to calculate area from only the three side lengths, without needing any angle. It's especially useful in surveying and land measurement.
Triangle solving is fundamental in navigation (triangulation), architecture and construction (roof trusses, load distribution), surveying (calculating land area), computer graphics (3D polygon meshes), and physics (vector decomposition).
The Law of Cosines is used when you know all three sides (SSS) or two sides and the included angle (SAS). It generalizes the Pythagorean theorem and allows you to find any angle or the third side of a triangle that is not necessarily right-angled.
Use Heron's Formula: first compute the semi-perimeter s = (a+b+c)/2, then calculate Area = √[s(s−a)(s−b)(s−c)]. Alternatively, if you know two sides and their included angle, use Area = (1/2)·b·c·sin(A).
No. Since the three angles must sum to 180°, a triangle can have at most one obtuse angle (greater than 90°). A triangle with two obtuse angles would require the angle sum to exceed 180°, which is impossible in Euclidean geometry.
The triangle inequality states that the sum of any two side lengths must be greater than the third side: a + b > c, a + c > b, and b + c > a. If this condition is violated, no valid triangle can be formed from those side lengths.
Simply subtract the sum of the two known angles from 180°. For example, if A = 50° and B = 70°, then C = 180° − 50° − 70° = 60°. This is the Angle Sum Property of triangles.
You need at least three elements with at least one side length. Valid combinations are SSS, SAS, ASA, AAS, and SSA. Providing only three angles (AAA) is not sufficient because it doesn't determine the triangle's size.
Yes. Enter a 90° angle for angle A (or B or C) and provide the two other known values. The calculator will apply the Law of Cosines or Law of Sines accordingly, and the result will satisfy the Pythagorean theorem.
In the SSA case, the given side opposite the known angle may be long enough to reach the base line at two different points, creating two geometrically valid triangles. The calculator will display both solutions and label them Triangle 1 and Triangle 2.
The calculator uses degrees by default, which is standard for most practical geometry problems. Ensure your angle inputs are in degrees unless you explicitly switch to radian mode.
Results are accurate to at least 10 significant figures internally and displayed to 2 decimal places by default. For most engineering and academic uses, this precision is more than sufficient.
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