Math
Triangle Calculator
Pick what you know — three sides, two sides and an angle, or two angles and a side — and this triangle calculator solves everything else: all sides and angles (degrees and radians), area, perimeter, semiperimeter, the three heights and medians, inradius, circumradius, and the triangle type.
SSA inputs are checked for the ambiguous case and both triangles are shown when two exist.
Opposite angle A.
Opposite angle B.
Opposite angle C.
Try an example
Result
Area
—
- Sides
- —
- Angles
- —
- Perimeter
- —
- Type
- —
Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
Use this calculator with
Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
What this triangle calculator solves
A triangle has six measurements: three sides (a, b, c) and three angles (A, B, C), where each side sits opposite the angle with the same letter. Any three of them, as long as at least one is a side, determine the triangle. The calculator picks the right tool for your combination: the law of cosines for SSS and SAS, the law of sines for ASA, AAS, and SSA, and Heron's formula for the area.
| Mode | You enter | First step used | Possible outcomes |
|---|---|---|---|
| SSS | a, b, c | Law of cosines for each angle | 1 triangle, or none if the triangle inequality fails |
| SAS | a, b, and included angle C | c² = a² + b² − 2ab·cos C | Always exactly 1 triangle |
| ASA | A, B, and included side c | C = 180° − A − B, then law of sines | 1 triangle, or none if A + B ≥ 180° |
| AAS | A, B, and side a | C = 180° − A − B, then law of sines | 1 triangle, or none if A + B ≥ 180° |
| SSA | a, b, and angle A | sin B = b·sin A / a | 0, 1, or 2 triangles (the ambiguous case) |
Formulas
The SSA ambiguous case
When you know two sides and an angle that is not between them, the law of sines gives sin B, and two angles share the same sine: B and 180° − B. Compare side a (opposite the known angle) with the altitude h = b·sin A. If a < h there is no triangle. If a = h the triangle is right-angled at B. If h < a < b two different triangles work — one with B acute and one with B obtuse. If a ≥ b only the acute solution fits. The calculator runs this check and shows Solution 1 and Solution 2 when both exist.
How to use it
- Choose the mode that matches the three measurements you know.
- Enter the sides and angles in the labeled boxes. Use the angle unit selector if your angles are in An angle unit where a full turn is 2π instead of 360 degrees..
- Read the solved sides and angles in the panel; each box says whether it was given or solved.
- Open Show the work to see the law of cosines or law of sines substitution written out.
- Use the derived values (heights, medians, inradius, circumradius) for follow-up questions without re-entering anything.
How to read the answer
- Angles are shown in degrees with radians underneath. The three always sum to 180° (π radians).
- The type line combines the side classification (equilateral, isosceles, scalene) with the angle classification (acute, right, obtuse).
- Height to a is the perpendicular distance from The turning point of a parabola, or a corner point of a shape. A to side a — the height you would pair with base a in Area = ½·base·height.
- The inradius is the radius of the circle that fits inside the triangle touching all three sides; the circumradius is the radius of the circle through the three vertices.
- The drawing is to scale, so a triangle that looks flat really is nearly flat.
Common mistakes and edge cases
- Triangle inequality: every pair of sides must add to more than the third side. 1, 2, 3 is not a triangle.
- Two angles that add to 180° or more leave nothing for the third angle — the calculator reports an error rather than a negative angle.
- A zero or negative side is rejected. Lengths must be positive.
- In SSA, if the side opposite the known angle is too short to reach the third side, no triangle exists; the error tells you the minimum length (b·sin A).
- Rounding: intermediate values are kept at full precision, and only the display is rounded. Rounding an angle to a whole degree before using the law of sines can shift a side noticeably.
Worked examples
SSS: sides 7, 10, 8
Three sides, solve the angles
Area
27.8107
SSS: 3-4-5 right triangle
Should classify as right
Area
6
SAS: 7 and 10 with 60° between
Law of cosines for the third side
Area
30.3109
ASA: 30°, 60°, included side 8
Third angle 90°, then law of sines
Area
13.8564
AAS: 30°, 60°, side a = 7
Two angles and a non-included side
Area
42.4352
SSA ambiguous: a = 7, b = 10, A = 30°
Two triangles fit — both are shown
Area
33.8981
SSA no solution: a = 3, b = 10, A = 30°
Side a is shorter than the altitude
Area
—
Radians: A = π/6, B = π/3, c = 8
Same ASA triangle entered in radians
Area
13.8564
Impossible sides 1, 2, 3
Triangle inequality fails
Area
—
Angles too large: 100° and 80°
No room for the third angle
Area
—
Decimal sides 2.5, 3.75, 4.1
Decimal inputs keep full precision
Area
4.605
Frequently asked questions
How do you solve a triangle with three sides?+
Use the law of cosines for one angle: cos A = (b² + c² − a²) / (2bc). Repeat for a second angle, then subtract both from 180° for the third. The area then comes from Heron's formula with s = (a + b + c)/2.
Why does SSA sometimes give two answers?+
The law of sines gives sin B, and both B and 180° − B have that sine. When side a is between the altitude b·sin A and side b, both angles produce a real triangle, so the calculator shows Solution 1 and Solution 2.
Can I solve a triangle from three angles (AAA)?+
No. Three angles fix the shape but not the size — every similar triangle shares those angles. You need at least one side. Use the similar triangles calculator if you know a scale factor.
What is the difference between ASA and AAS?+
In ASA the known side is between the two known angles; in AAS it is not. Both are solved the same way: find the third angle from the angle sum, then use the law of sines.
How do I know if a triangle is acute, right, or obtuse from the 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 does this comparison through the largest angle.
Does this calculator work in radians?+
Yes. Set the angle unit to radians for your inputs. Results always show degrees with the radian value underneath.
What are the heights and medians for?+
Each height is the perpendicular from a vertex to the opposite side, which you need for Area = ½·base·height. A median joins a vertex to the midpoint of the opposite side; the three medians meet at the centroid.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026