About the Triangle Calculator
This triangle calculator solves a complete triangle from any three values that determine it. Pick the combination you know — three sides (SSS), two sides and the angle between them (SAS), two angles and the side between them (ASA), two angles and a non-included side (AAS), or two sides and a non-included angle (SSA) — and it returns every side, every angle, the area, the perimeter and whether the triangle is acute, right or obtuse, scalene, isosceles or equilateral.
It is built for trigonometry and geometry students checking law-of-sines and law-of-cosines working, and for practical tasks such as surveying a triangular plot, laying out roof trusses or cutting a triangular panel where not every side can be measured directly.
Side a is opposite angle A, b opposite B and c opposite C. Angles are in degrees. SSA is the “ambiguous case”: some inputs fit two different triangles, in which case the calculator shows the one with the acute angle B and mentions the other in the notes.
With the default inputs, the area is 26.8328. Change any value above to recalculate instantly.
How to use the triangle calculator
- 1Choose which three values you know (SSS, SAS, ASA, AAS or SSA).
- 2Enter the sides and angles shown — angles in degrees.
- 3Make sure each side label matches the angle opposite it.
- 4Read the missing sides and angles, area, perimeter and type.
- 5For SSA, check the notes for a possible second solution.
Formula and method
The calculator first finds all three sides. With SAS it uses the law of cosines to get the third side; with ASA and AAS the third angle is 180° minus the other two, and the law of sines scales the known side to the others; with SSA the law of sines gives angle B, checking whether zero, one or two triangles are possible.
Once the sides are known, every angle is recomputed with the law of cosines so the results are consistent, and the area comes from Heron’s formula using the semi-perimeter s = (a + b + c) ÷ 2. The inradius is area ÷ s, the circumradius is abc ÷ (4 × area), and the height onto side a is 2 × area ÷ a.
- a, b, c
- Side lengths
- A, B, C
- Angles opposite sides a, b and c (degrees)
- s
- Semi-perimeter, (a + b + c) ÷ 2
- r, R
- Radius of the inscribed and circumscribed circles
Worked examples
Three sides: 7, 8 and 9 (SSS)
The semi-perimeter is 12, so Heron gives √(12 × 5 × 4 × 3) = √720 ≈ 26.83. The law of cosines gives angles of about 48.19°, 58.41° and 73.40° — all under 90°, so the triangle is acute.
Two sides 10 and 7 with a 60° angle between them (SAS)
c² = 100 + 49 − 2 × 10 × 7 × cos 60° = 79, so c ≈ 8.89. The area is ½ × 10 × 7 × sin 60° ≈ 30.31.
Angles 40° and 60° with side c = 12 between them (ASA)
The third angle is 180 − 40 − 60 = 80°. By the law of sines a = 12 × sin 40° ÷ sin 80° ≈ 7.83 and b = 12 × sin 60° ÷ sin 80° ≈ 10.55.
Ambiguous case: a = 10, b = 12, A = 40° (SSA)
sin B = 12 × sin 40° ÷ 10 ≈ 0.771, so B ≈ 50.47° or 129.53°. Both work here; the acute solution gives C ≈ 89.53° and c ≈ 15.56.
Frequently asked questions
How many values do I need to solve a triangle?+
Three, and at least one must be a side. Three angles alone fix the shape but not the size, so the triangle cannot be solved without a length.
When do I use the law of sines vs the law of cosines?+
Use the law of cosines when you know three sides (SSS) or two sides and the included angle (SAS). Use the law of sines when you know two angles and a side (ASA, AAS) or in the SSA case.
What is the ambiguous case (SSA)?+
When you know two sides and an angle that is not between them, the given side can swing to meet the base in two places, one place, or not at all. That can produce two valid triangles, one, or none.
How do I find the area of a triangle from three sides?+
Use Heron’s formula: compute s = (a + b + c) ÷ 2, then area = √(s(s − a)(s − b)(s − c)). For sides 7, 8 and 9, s = 12 and the area is √720 ≈ 26.83.
What is the triangle inequality?+
Each side must be shorter than the sum of the other two. Sides 3, 4 and 8 cannot form a triangle because 3 + 4 is less than 8.