Which three facts actually determine a triangle
A triangle has six parts, three sides and three angles, and it takes three of them to fix the shape — but not any three. Three sides work. Two sides with the angle caught between them work. Two angles with any side work, because the third angle is forced and the law of sines scales the rest. Three angles do not: they fix the shape but not the size, so you get a family of similar triangles rather than one answer.
The awkward case is two sides and an angle that is not between them. That combination can describe two different triangles, one acute and one obtuse, both perfectly valid. Textbooks call it the ambiguous case. Rather than pick one and hope, this calculator leaves the case out. If that is what you have measured, measure one more thing — the third side is usually easiest — and use the three-sides mode.
The formulas being used
- Triangle inequality: any two sides must add to more than the third. 1, 2 and 10 fails, and no amount of rearranging helps: the short pair cannot span the gap.
- Heron's formula: with s as half the perimeter, area = sqrt(s(s-a)(s-b)(s-c)). It gives area from three sides with no height needed. This page uses Kahan's rearranged version of it, which matters when a triangle is long and thin and the plain form throws away significant digits.
- Law of cosines: c² = a² + b² - 2ab·cos C. Run forwards it gives a side from two sides and the included angle; rearranged as cos C = (a² + b² - c²) / 2ab it gives every angle from three sides.
- Law of sines: a / sin A = b / sin B = c / sin C. This is what solves the two-angles-and-a-side case.
- Pythagoras: the law of cosines when C is 90 degrees and the cosine term vanishes.
- Height: area = base × height / 2, so the height onto any side is twice the area divided by that side.
Reading the classification
By angle, the largest angle decides: under 90 degrees acute, exactly 90 right, over 90 obtuse. You can see which without computing any angle at all — square the longest side and compare it to the sum of the other two squared. Larger means obtuse, equal means right, smaller means acute. By sides, two equal makes it isosceles and three equal makes it equilateral, and an equilateral triangle is also isosceles.
The right-triangle check is the one that earns its keep on site. Measure 3 feet along one wall, 4 feet along the other, and if the diagonal between those marks is exactly 5 feet the corner is square. Any multiple works and bigger is better: 6-8-10 or 12-16-20 spreads the same check over a longer run, so a given error in your marks translates into a smaller error in the angle.
A worked example, and where it goes wrong
Sides 5, 6 and 7. The semi-perimeter is 9, so the area is sqrt(9 × 4 × 3 × 2) = sqrt(216), about 14.697. The largest angle faces the 7 side: cos C = (25 + 36 - 49) / 60 = 0.2, so C is about 78.46 degrees, and the triangle is acute and scalene. Height onto the 5 side is 2 × 14.697 / 5, about 5.879.
Now take sides 100, 100 and 0.5 — a real shape, a very thin wedge. The area is about 24.998, and the two big angles are each about 89.86 degrees. Move the short side by a hundredth and both angles shift noticeably. That sensitivity is not a defect in the arithmetic, it is the geometry: near-degenerate triangles convert small length errors into large angle errors, and no calculator can give you angles more precise than your tape measure allows. When the answer matters and the triangle is thin, measure the angle directly instead of deriving it.
Questions people ask
Why does 1, 2, 10 give an error instead of a number?
Because there is no such triangle. Lay a 1 unit stick and a 2 unit stick end to end and you have covered 3 units; the 10 unit side leaves a gap of 7 that nothing can bridge. Heron's formula would happily return the square root of a negative number here, which is where a naive calculator produces NaN or a nonsense figure. The check is the triangle inequality: every pair of sides must add to more than the remaining one, and it must hold for all three pairings. If your measurements fail it, one of them is wrong — most often a side was measured along a different path than you think, over a curve or a jog rather than straight corner to corner.
What units does this use?
Whatever you enter, as long as all the lengths share it. The unit selector only changes labels and adds a couple of convenient area conversions. Enter inches and the area comes out in square inches, with the square feet shown alongside since that is usually what you want for material. Angles are always in degrees, never radians. If your source gives radians, multiply by 57.29578 first, or use the angle converter on this site.
How do I find the hypotenuse if I know one leg and the hypotenuse?
Pick the right-triangle mode, put the hypotenuse in the c box and the known leg in a or b, and clear the box for the side you want. It solves with the square root of the difference of squares. If you fill all three boxes it will assume a and b are the legs, recompute the hypotenuse from them, and tell you it did so — that way a typo shows up rather than quietly producing a triangle you did not measure.
Can I get the angles without knowing all three sides?
Only if what you know is enough to fix the shape. Two sides and the angle between them is enough, and the calculator returns all three angles. Two angles and any side is enough, and the third angle is just 180 minus the other two. A base and a height is not enough for angles, because the apex can slide sideways without changing either, and that mode deliberately returns area only rather than inventing a shape.