Triangle Calculator
Solve triangle problems with right triangles, SSS, SAS, ASA/AAS, SSA ambiguous cases, area formulas, and triangle type classification using step-by-step explanations and visual diagrams.
Background
Triangle problems usually come down to what you know: sides, angles, height, or a special right-triangle relationship. This calculator helps students choose the right method, solve missing values, and understand the geometry visually.
How to use this calculator
- Choose the mode that matches what you know: right triangle, SSS, SAS, ASA/AAS, SSA, area, or classify.
- Enter the given sides and angles. Angles are entered in degrees.
- Use Help Me Choose if you are not sure which formula applies.
- Click Calculate to see missing sides, missing angles, area, perimeter, triangle type, a visual diagram, and steps.
How this calculator works
- Right triangles use the Pythagorean theorem and trig ratios.
- SSS uses Law of Cosines to find angles, then Heron’s formula for area.
- SAS uses Law of Cosines to find the missing side, then Law of Sines or Cosines for angles.
- ASA/AAS uses the triangle angle sum and Law of Sines.
- SSA checks the ambiguous case, where there may be no triangle, one triangle, or two triangles.
- Area mode supports base-height, Heron’s formula, and the sine area formula.
Formula & Equations Used
Angle sum: A + B + C = 180°
Perimeter / total boundary length: P = a + b + c
Pythagorean theorem: a² + b² = c²
Law of Sines: a / sin A = b / sin B = c / sin C
Law of Cosines: c² = a² + b² − 2ab cos C
Area, base-height: Area = ½bh
Area, SAS: Area = ½ab sin C
Heron’s formula: Area = √[s(s−a)(s−b)(s−c)]
Height (altitude) to a side: h = 2 × Area ÷ side
Example Problems & Step-by-Step Solutions
Example 1 — Right triangle with legs 3 and 4
- Use a² + b² = c².
- 3² + 4² = 9 + 16 = 25.
- c = √25 = 5.
Example 2 — Area from two sides and included angle
- Use Area = ½ab sin C.
- If a = 8, b = 5, and C = 60°, then Area = ½(8)(5)sin60°.
- The calculator shows the substitution, result, and triangle diagram.
Example 3 — SSA ambiguous case
- SSA gives two sides and a non-included angle.
- Use Law of Sines to test possible angle values.
- The calculator reports whether 0, 1, or 2 triangles are possible.
Example 4 — Finding a missing height
- If you know the base and the area, height is h = 2 × Area ÷ base.
- Enter any two of base, height, and area in Area mode and the calculator solves the third.
- Every solved triangle also reports the altitude to each of its three sides.
Frequently Asked Questions
Q: What can this Triangle Calculator solve?
It can solve right triangles, SSS, SAS, ASA/AAS, SSA ambiguous cases, triangle area, perimeter, height (altitude), and triangle type classification.
Q: When should I use Law of Sines?
Use Law of Sines when you have a known side-angle pair, especially ASA, AAS, and many SSA problems.
Q: When should I use Law of Cosines?
Use Law of Cosines for SSS or SAS problems, especially when you do not yet have a side-angle pair.
Q: What is the SSA ambiguous case?
SSA can sometimes create no triangle, one triangle, or two possible triangles. This calculator checks that case directly.
Q: Can this calculator classify a triangle?
Yes. It can classify by side lengths and by angles, such as scalene, isosceles, equilateral, acute, right, or obtuse.
Q: Can this calculator find the height of a triangle?
Yes. Every solved triangle shows the altitude (height) to each of its three sides, calculated from the area. You can also enter any two of base, height, and area in Area mode to solve for the third.
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