What this calculator solves
Any triangle is pinned down completely by just three of its six parts, sides a, b, c and the angles A, B, C opposite them, provided one of the three is a side. Give this calculator any such combination and it fills in the rest: every missing side and angle, the area, perimeter, the height from each vertex, the inscribed and circumscribed circle radii, and a plain classification of the shape. Nothing here assumes a 90° angle anywhere, so the maths covers acute, right and obtuse triangles alike, which is why this page exists separately from the site's right-triangle-only tool.
Which method fires depends entirely on which three parts are supplied, detected automatically. Three sides, or two sides with the angle between them, go through the law of cosines. Two angles plus any one side go through the law of sines. Two sides plus an angle that ISN'T between them is the tricky combination, since it can produce no triangle, one, or two entirely different ones; that ambiguous case gets its own explanation below. If the triangle is already known to be right-angled, the dedicated Right Triangle Calculator is a more streamlined tool for that one shape.
The law of cosines: solving from three sides, or two sides and the angle between them
The law of cosines generalises Pythagoras' theorem to triangles that aren't right-angled, folding in a correction term for whatever the angle between two sides actually is.
Given two sides and the included angle (SAS), this calculator applies the formula once to find the third side, then twice more, now every side is known, to get the remaining angles. Given all three sides (SSS), it rearranges the formula to solve for cosine: cos A = (b² + c² − a²) / (2bc), and the same for B, with C found by subtracting from 180°. Take sides 7, 10 and 5: cos A = (100 + 25 − 49) / 100 = 0.76, so A = arccos(0.76) ≈ 40.54°; the same approach gives B ≈ 111.80°, leaving C ≈ 27.66°. B is obtuse here, and that's the point: the law of cosines handles obtuse angles just as naturally as acute ones, unlike the SOH-CAH-TOA ratios used for right triangles.
The law of sines: solving from two angles and any one side
Once two of a triangle's angles are known, the third follows immediately, since all three always add up to 180°. What the law of sines adds is a way to scale actual side lengths once you know one of them, using the fact that every side is proportional to the sine of its opposite angle.
Take angle A = 50°, side c = 8, and angle B = 60°. The third angle follows straight away: C = 180° − 50° − 60° = 70°. The shared ratio is c / sin C = 8 / sin(70°) ≈ 8.515, and multiplying that by sin(50°) and sin(60°) gives a ≈ 6.52 and b ≈ 7.38. This works identically whichever side is known, so this calculator treats what's traditionally taught as two cases, ASA (the side sits between the two known angles) and AAS (it doesn't), as one and the same: once two angles are known the third is fixed regardless, and the law of sines doesn't care which side was measured.
The ambiguous case: why SSA can give one, two, or no triangles
Two sides and an angle that ISN'T between them, often written SSA, is the one combination that doesn't always pin down a single triangle. Picture side b fixed at one end, angle A hinged at that end, and side a a rigid rod free to swing from the far end of b down to the base line. Too short, and it never reaches the base: no triangle. Just long enough to touch at a right angle: one triangle. Longer than that but still shorter than b, and it can swing to two positions and still reach the base, giving two different triangles from the same three numbers. At least as long as b, and only one position is possible, so there's exactly one triangle again.
Take a = 7, b = 10, A = 40°. The swing height is h = 10 × sin(40°) ≈ 6.43, and 7 sits between that and 10, so two triangles exist. The law of sines gives sin B = b sin A / a ≈ 0.918, and arcsin of that has two valid answers: B ≈ 66.67° or its supplement, B ≈ 113.33°. Each leads to a different third angle and side, so this calculator shows both triangles side by side rather than silently picking one. Change the first side to a = 3, keeping b = 10 and A = 40°, and a is now shorter than h ≈ 6.43: no triangle fits those numbers, and this calculator says so rather than forcing an answer.
Why three angles alone aren't enough
Enter angle A, angle B and angle C with no side at all and this calculator declines to solve anything, on purpose. Three angles fix a triangle's proportions but say nothing about its actual size: a tiny triangle and a huge one can share exactly the same three angles (similar triangles), since scaling every side by the same factor never changes any angle. At least one side length has to be added before any side, the area or the perimeter can be calculated.
Area, perimeter and the three heights
The area is always shown worked out two independent ways, as a check that both agree. The first is Heron's formula, needing only the three side lengths, named after the Alexandrian mathematician and engineer Heron (sometimes Hero), who described it roughly two thousand years ago.
For sides 7, 10 and 5, the semi-perimeter is s = 11, and the area works out to √(11 × 4 × 1 × 6) = √264 ≈ 16.25. The second method uses any two sides and the angle between them directly: Area = ½ × a × b × sin C. Both formulas describe the same triangle, so they agree once rounding is accounted for. The perimeter is simply a + b + c. Each of the three heights, the perpendicular distance from a vertex to the opposite side, comes from rearranging the area formula: height from side a = 2 × Area ÷ a, and the same pattern for b and c. A longer side always carries a shorter matching height, since the area they multiply to give stays fixed.
The inscribed and circumscribed circles
Every triangle has exactly one circle touching all three sides from the inside (the incircle) and one circle passing through all three corners (the circumcircle), with radii that follow from the sides and area already found: r = Area / s, and R = (a × b × c) / (4 × Area). For the 7-10-5 triangle, r = 16.25 / 11 ≈ 1.477 and R = 350 / 65 ≈ 5.385. The inradius is useful for checking a triangular space against a round object; the circumradius answers the reverse question, how large a circle is needed to enclose the triangle entirely.
Classifying the triangle
Alongside the numbers, this calculator names the shape in everyday terms: acute if all three angles are under 90°, right if one is exactly 90°, or obtuse if one is over 90° (only one of the three can ever apply); and equilateral if all three sides match, isosceles if exactly two do, or scalene if all three differ. An equilateral triangle is always acute (60° at every corner), but isosceles and scalene triangles can be acute, right or obtuse.
Where general triangle solving shows up: surveying, navigation, construction
Surveyors measure a baseline and two angles to a distant point (triangulation), then use this maths to work out distances they can't measure directly. Ships and aircraft use the law of sines and cosines to turn a bearing and a known distance into a position fix. Builders and engineers rely on it for anything triangular that isn't a right angle, from an unusual roof pitch to checking that fixed anchor points will take the load a design assumes.
Reading the results panel
The headline is the area, alongside a summary of every side, angle and the shape classification. Below sits a note naming which law fired, the full worked substitution, the triangle drawn to scale with each vertex marked in a different colour, and two tables: every side and angle, then the perimeter, semi-perimeter, both area formulas, the three heights, the inradius and the circumradius. On the rare SSA input where two triangles fit, those blocks are replaced by two matching panels side by side, one per solution.
Questions people ask
Which three values do I actually need to know?
Any three of the six: sides a, b, c and angles A, B, C, as long as one is a side. Three angles alone only fix the shape, not the size, so they aren't enough on their own.
How does this calculator know which method to use?
It looks at exactly which three fields are filled in. Three sides, or two sides with the angle between them, use the law of cosines. Two angles with any one side use the law of sines. Two sides with a non-included angle is checked for the ambiguous SSA case, which can have zero, one or two solutions.
Why did I get two different triangles back?
Two sides and a non-included angle (SSA) were entered, and the numbers fall in the range where two genuinely different triangles both fit: one with an acute angle at the third vertex, one with its obtuse supplement. Both are shown side by side rather than one being silently chosen.
Why does it sometimes say no triangle exists at all?
This happens with the same SSA combination, when the side opposite the known angle is too short to ever reach the third side, however it's swung. The calculator checks this directly rather than returning a wrong result.
Why are two area formulas shown?
Heron's formula uses only the three sides; ½ab sin C uses two sides and the angle between them. Both describe the same triangle, so showing both is a built-in consistency check.
Does this work for right triangles too?
Yes, a right triangle is simply one where an angle happens to be 90°, classified as such automatically. If it's already known to be right-angled, the dedicated Right Triangle Calculator is quicker.
Can I enter angles in radians instead of degrees?
Yes, the Degrees/Radians toggle next to the angle fields changes what the boxes expect and display; the solved triangle stays identical either way, since it's only a different way of measuring the same angles.
This calculator solves any valid triangle from any three of its six parts. It won't invent a triangle from an invalid combination: side lengths that fail the triangle inequality, angles that don't leave room for a third, or three angles with no side are all explained rather than forced into a number.