Any three measurements — and both answers when there are two.
Any three measurements
Angle A is opposite side a, and so on.
solved from two sides and a non-included angle (SSA)
two triangles fit
Your measurements describe two different triangles. Both are shown below — the second is not an approximation of the first.
Area
79.4256
by the stable Heron form
Perimeter
45.8564
semiperimeter 22.9282
Inradius
3.4641
the inscribed circle
Circumradius
10
the circle through all three corners
Classification
obtuse and scalene
by angles and by sides
Angles sum to
180°
always 180, and worth checking
First triangle
Each side with the angle opposite it, and the height dropped to that side
Side
Opposite angle
Height to it
a
10
30°
15.8851
b
16
53.1301°
9.9282
c
19.8564
96.8699°
8
Second triangle — equally valid
Each side with the angle opposite it, and the height dropped to that side
Side
Opposite angle
Height to it
a
10
30°
6.28513
b
16
126.87°
3.9282
c
7.85641
23.1301°
8
Second area
31.4256
the other solution
Second perimeter
33.8564
also consistent with your input
Second classification
obtuse and scalene
a genuinely different shape
Which is yours
not decidable
the measurements alone cannot say
This is the AMBIGUOUS case. Two sides and an angle NOT between them do not always fix one triangle: the sine rule gives an angle, and 180° minus that angle has the same sine.
Both are real triangles with your measurements. The second has its second angle at 126.870° instead of 53.1301°, and it is shown below — a solver that returned only one of these would be hiding a correct answer.
If you know the triangle is acute, or that the unknown angle is obtuse, that extra fact picks one. The measurements alone do not.
The area uses Heron’s stable form, which survives a needle-shaped triangle the textbook version returns NaN for.
What this tool shows
Two sides and the angle between them fix one triangle. Two sides and an angle NOT between them can fix two, and most solvers quietly return one. This page returns both, because the second is not an approximation of the first — it is a different triangle with your measurements.
A triangle solved from any three measurements
Both solutions when SSA is ambiguous
Area, by the numerically stable Heron form
The height to each side
Inradius and circumradius
Why some measurements describe no triangle at all
Every case: SSS to SSA Sine rule verified Both ambiguous answers Area, heights and radii
Updated 7 September 2026 · Works in any browser, no installation
Three measurements fix a triangle — except in one case. Three sides, two sides with the angle between them, or one side with two angles all give exactly one answer. Two sides with an angle NOT between them can give two, one, or none, and which of those depends on the numbers rather than on the shape of the input.
At a glance
Formula shown
The cosine rule c\u00b2 = a\u00b2 + b\u00b2 \u2212 2ab\u00b7cos C solves SSS and SAS. The sine rule a/sin A = b/sin B = c/sin C solves ASA, AAS and SSA \u2014 and in SSA it gives two candidate angles, since sin \u03b8 and sin(180\u00b0 \u2212 \u03b8) are equal.
Scenario support
A surveying or navigation triangle; a geometry exercise with three given measurements; checking whether a set of measurements is consistent at all.
Educational estimate
Planning support from the values you enter — not professional advice.
Which case is which
SSS — three sides. One triangle, always, provided the two shorter sides reach further than the longest. The cosine rule gives every angle.
SAS — two sides and the angle BETWEEN them. One triangle. The cosine rule gives the third side directly.
ASA and AAS — one side and two angles. One triangle. The third angle follows from the 180° sum, and the sine rule scales the remaining sides.
SSA — two sides and an angle NOT between them. Two, one or none. This is the ambiguous case, and it is the only one that needs care.
AAA — three angles. Infinitely many triangles, all the same shape and any size. The page refuses rather than picking one.
The ambiguous case
Take a = 10, A = 30° and b = 16. The sine rule gives sin B = 0.8, so B is 53.13°. It is also 126.87°, because sin(180° − θ) equals sin θ.
Both work. One triangle is acute at B, the other obtuse, and both have a side of 10 opposite a 30° angle and a second side of 16. Nothing in the measurements distinguishes them.
Geometrically: swing a side of length 10 from the end of the 16, and it can meet the base line at two different points. The two intersections are the two triangles.
Three outcomes. If the swinging side is shorter than the perpendicular distance, it never reaches and there is no triangle. If it is exactly that length it touches once, giving one right-angled triangle. If it is longer it crosses twice — unless it is longer than the fixed side too, in which case one of the two crossings falls on the wrong side and only one triangle survives.
The page reports which of those you have, and shows both triangles when there are two. Choosing between them needs a fact the measurements do not contain: that the triangle is acute, say, or that a particular angle is obtuse.
Sine rule or cosine rule
Cosine rule when you have three sides, or two sides and the angle between them. It is c² = a² + b² − 2ab·cos C, and it is Pythagoras with a correction term that vanishes exactly when C is 90°.
Sine rule when you have an angle and the side opposite it, plus one more thing. a/sin A = b/sin B = c/sin C, and that common value is twice the circumradius.
A practical point: prefer the cosine rule for finding an ANGLE when you can. Inverse cosine distinguishes acute from obtuse on its own, while inverse sine does not — which is exactly what makes SSA ambiguous.
When no triangle exists
The triangle inequality. The two shorter sides must together exceed the longest. 1, 2 and 5 fail: 1 + 2 = 3 never reaches 5, so the sides cannot close.
Exactly equal is still no triangle. 1, 2 and 3 lie flat — a straight line with zero area. It is the boundary of the inequality rather than a triangle.
Angles that overrun. Two angles totalling 180° or more leave nothing for the third.
SSA that cannot reach. With a = 5, A = 60° and b = 20, the side of length 5 would have to be at least 17.3 to reach the base line. The sine rule asks for a sine above 1, which is the arithmetic saying the same thing.
Area without a height
Half base times height needs a height, and a height is usually the thing you do not have.
From three sides: Heron’s formula. No angle, no height, just the three lengths.
From two sides and the angle between: ½ab·sin C. Often the quickest route, and the one the suite checks Heron against.
Once the area is known, every height follows: h = 2 × area ÷ base, for each side in turn. The page reports all three.
It uses Heron in the numerically stable rearrangement rather than the textbook one, because on a long thin triangle the textbook version loses most of its significant digits — and the area is what the heights are derived from, so the error propagates.
The two circles
Every triangle has exactly one circle inside it touching all three sides, and exactly one outside passing through all three corners.
Inradius r = area ÷ s, where s is the semiperimeter. The incircle’s centre is where the angle bisectors meet.
Circumradius R = abc ÷ (4 × area). Its centre is where the perpendicular bisectors of the sides meet, and for an obtuse triangle that point lies OUTSIDE the triangle.
The sine rule’s common value is 2R, which is why a/sin A gives the circumdiameter directly — a fact the page uses as one of its checks.
Sources and methodology
The solution cases and the ambiguity are standard; these are the references.
Method. SSS and SAS go through the cosine rule, ASA and AAS through the sine rule, and SSA through the sine rule with BOTH candidate angles kept. The boundary — where the sine works out to 1 and the two candidates coincide at a right angle — is tested on the sine rather than on the angle, because asin is ill-conditioned near 1 and testing afterwards presented one right triangle as an ambiguous pair. Areas use Heron’s stable rearrangement, and the suite verifies every solved triangle against the sine rule and against ½ab·sin C, neither of which the solver uses for SSS. That engine is verified on every change against 76 hand-written assertions, including that the sine rule holds and the angles sum to 180° across four thousand solved triangles, and that the area agrees with ½ab·sin C — a route the solver does not take. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Pythagorean Theorema² + b² = c² solved for whichever side you leave blank, with the triangle drawn to scale and integer triples reduced to their primitive.
Heron’s FormulaArea from three sides in the rearrangement that keeps its digits — with the textbook form shown beside it so you can see what the usual order costs.
Special Right Triangles30-60-90 and 45-45-90 from any one side, in exact surd form as well as decimals — with the warning that the long leg is √3 times the short, not twice.
Similar TrianglesScale factor and the missing sides — plus the area factor, which is its square. Two pairs with different ratios are reported as not similar rather than averaged.
Reference AngleThe acute angle to the x-axis — never the y-axis — with the quadrant rule written out using your own numbers and the sign of each trig function beside it.
Angle Between Two VectorsDot product, magnitudes and the angle as separate steps, in the plane or in space — with exactly parallel vectors returning 0° rather than a floating-point smudge.
For a right triangle specifically, the Pythagorean Theorem Calculator is quicker, and the Special Right Triangles Calculator gives exact surd answers for the 30-60-90 and 45-45-90 shapes.
Educational use disclaimer
This is an educational tool. The arithmetic is exact to double precision; whether a set of measurements describes the triangle you meant is a modelling question, and in the ambiguous case the measurements alone cannot answer it.
Published the triangle solver returning BOTH answers in the ambiguous SSA case, since two sides and a non-included angle can describe two triangles and returning one of them hides a correct answer.
The SSA boundary is detected on the sine rather than on the angle: asin is ill-conditioned near 1, and testing afterwards made one right triangle look like an ambiguous pair. The suite caught that.
Areas use Heron's numerically stable rearrangement, because every reported height is derived from the area and inherits its error.
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