How to use the Triangle Calculator
- Choose what you know: three sides (SSS), two sides and the angle between them (SAS), two angles and a side (ASA or AAS), two sides and a non-included angle (SSA), or two values of a right triangle.
- Enter the values. Side a is opposite angle A, side b opposite B and side c opposite C — the diagram shows which is which.
- Read the missing sides and angles, area, perimeter, heights, medians, inradius and circumradius. The working shows each step.
What does this tool do?
Three independent measurements fix a triangle, as long as one of them is a side. The calculator applies the law of cosines when you know sides around an angle, the law of sines when you know angles and their opposite sides, and Heron's formula for the area, then draws the triangle to scale with the values you entered highlighted.
The SSA case is handled properly. Two sides and an angle that isn't between them can describe no triangle, exactly one, or two different triangles — the “ambiguous case”. The calculator works out which, explains why using the height h = b·sin A, and shows both triangles side by side when there are two.
For right triangles, enter any two of the legs, the hypotenuse and the acute angles, and Pythagoras and the basic trig ratios fill in the rest. Angles can be entered and shown in degrees or radians, with up to eight decimal places.
Why use it?
- Six ways to solve: SSS, SAS, ASA, AAS, SSA and right triangles.
- Finds both answers in the SSA ambiguous case — and says when no triangle exists.
- Scaled diagram with the given values highlighted.
- Area, perimeter, heights, medians, inradius and circumradius.
- Step-by-step working with the law of sines and cosines.
Use cases
- Checking trigonometry homework and exam practice answers.
- Roof pitches, rafters, ramps and stair stringers in DIY and construction.
- Surveying and navigation: distances from two bearings and a baseline.
- Woodwork and design: cutting angles for frames and triangular panels.
Example: the ambiguous case
Side a = 7, side b = 10 and angle A = 40°. The height from C is h = 10 × sin 40° ≈ 6.43. Because h < a < b, side a can meet the base in two places:
| Triangle 1 | Triangle 2 | |
|---|---|---|
| Angle B | 66.67° | 113.33° |
| Angle C | 73.33° | 26.67° |
| Side c | 10.43 | 4.89 |
| Area | 33.53 | 15.71 |
Common mistakes
- Mixing up which side is opposite which angle. Side a must be across from angle A; using the adjacent side gives a different triangle.
- Stopping at one answer in an SSA problem. sin⁻¹ returns only the acute angle; its supplement (180° minus it) may give a second valid triangle.
- Entering degrees while the calculator (or a scientific calculator) is set to radians — 40 radians is about 2,292°.
- Using sides that break the triangle inequality, such as 2, 3 and 6: the two shorter sides must add up to more than the longest.
The formula
Privacy
The calculation happens instantly in your browser. The numbers you enter are not sent to our servers or saved. There's no account to create and nothing to install.
Frequently asked questions
How do I find the missing side of a triangle?
With two sides and the angle between them, use the law of cosines (SAS). With two angles and any side, use the law of sines (ASA or AAS). For a right triangle with two sides known, use Pythagoras. Choose the matching option and the calculator does the rest.
What is the ambiguous case of the law of sines?
When you know two sides and an angle that isn't between them (SSA), the information can fit two different triangles, one, or none. It happens because sin θ = sin(180° − θ), so an angle found with the sine rule may be acute or obtuse.
How do I find the area of a triangle from three sides?
Use Heron's formula: work out 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.
Can I solve a triangle from three angles?
No — three angles fix the shape but not the size, so infinitely many similar triangles fit. You need at least one side length.
How do I tell if a triangle is right, acute or obtuse from its sides?
Compare the square of the longest side with the sum of the squares of the other two: equal means right, smaller means acute, larger means obtuse. The calculator classifies every solution.
Last reviewed by the M2Toolkit team.