Enter exactly three values. Use one unit for all sides; angles are in degrees.
Solved triangle
Triangle 1 · labels show the solved measurements. The sketch is not to scale.
Text alternative: Triangle 1: sides a 3, b 4, c 5; angles A 36.87 degrees, B 53.13 degrees, C 90 degrees. The drawing is normalised and not to scale.
Side a
3
Side b
4
Side c
5
Angle A
36.8699°
Angle B
53.1301°
Angle C
90°
Area (Heron)
6 square units
Area (½ab sin C)
6 square units
Perimeter
12 units
Semi-perimeter
6 units
Inradius
1 units
Circumradius
2.5 units
Height to side a
4 units
Height to side b
3 units
Height to side c
2.4 units
Median to side a
4.272 units
Median to side b
3.6056 units
Median to side c
2.5 units
Side type
scalene
Angle type
right
Lengths use your chosen unit; areas use that unit squared. Three angles alone do not set a triangle’s size. The diagram is normalised and not to scale.
Three matching measurements determine the rest of a triangle. For the default 3–4–5 example, the calculator finds three angles and area 6 square units.
Step 1Choose a known caseSSS · SAS · ASA/AAS · SSAEnter exactly three measurements, including at least one side.
Step 2Apply a triangle ruleSine or cosine ruleThe angle-side pairing decides which rule can solve the missing values.
Step 3Check the geometry36.87° + 53.13° + 90° = 180°The result also checks area by two formulas.
The letters follow a fixed convention: lowercase side a is opposite uppercase angle A, and likewise for b/B and c/C. Lengths can use any unit, but all entered sides must use the same one.
The main solve cases
SSS, SAS, ASA or AASOne triangleThree sides; two sides and their included angle; or two angles plus one side.
SSA2 possible trianglesThe sine rule can produce an angle and its supplement. The calculator displays both when both fit.
Three angles alone reveal a shape, not its scale, so at least one side is required.
Worked example
A triangle with sides 3, 4 and 5 has perimeter 12 units and area 6 square units.
Solving the 3–4–5 triangle
Semi-perimeter
s = (a + b + c) ÷ 2
Heron area
K = √[s(s − a)(s − b)(s − c)]
Angle check
A + B + C = 180°
a, b, cthe three side lengths 3, 4, 5
shalf the perimeter 6
Ktriangle area 6 square units
Perimeter3 + 4 + 512 units
Semi-perimeter12 ÷ 26 units
Heron area√(6 × 3 × 2 × 1)6 square units
Sine area check½ × a × b × sin C6 square units
The area values agree because both formulas describe the same triangle. The calculator keeps full precision internally and rounds only the displayed values.
Reading your result
The default triangle is scalene and right; each solved value is tied to the letter printed on the diagram.
Sides and anglesAll three side lengths and their opposite angles.
Area and perimeterArea by Heron and by ½ab sin C, plus perimeter and semi-perimeter.
Radii, heights and mediansThe inradius, circumradius, perpendicular heights and vertex-to-midpoint medians.
Triangle typeSide class and angle class, with documented numeric tolerances.
A height measures a perpendicular drop to the opposite side; a median reaches that side’s midpoint. The SVG sketch is labelled and accessible, but is normalised to fit and is not a scale drawing.
Why SSA may show two results
The SSA example a=6, b=8, A=30° has 2 valid branches; both are shown so you can compare their angles.
The sine rule can identify two angles with the same sine: an acute angle and its supplement. Each candidate is kept only when the remaining angle is positive. A branch that fails that check is not a triangle.
Assumptions and limits
This tool solves flat, Euclidean triangles from consistent measurements; it does not guess a unit or fit conflicting extra data.
Spherical geometryThe rules assume a flat plane, not triangles drawn on a globe.
Named unitsThe tool cannot know whether a side is in centimetres, metres or another unit.
Extra measurementsExactly three values are accepted; inconsistent redundant measurements are not averaged.
Angle classification treats values within 1e−9° of a right angle as right. Side classification treats lengths within 1e−9 times the largest side as equal. These tolerances change the type label only.
Related geometry tools
A triangle is often one part of a bigger shape problem. Other tools cover the pieces around it.
For a sloped roof, the roofing calculator turns a footprint and pitch into rafter length and area.
Common mistakes
Match each side to its opposite angle, and do not mix length units.
Mistake: Entering A as the angle between b and c.
Instead: A is opposite a; the angle between b and c is A.
Mistake: Using three angles and expecting a unique size.
Instead: Add one side; the angles alone only set the shape.
Mistake: Treating SSA as always having one answer.
Instead: Check whether the page shows both valid triangles.
Mistake: Reading the SVG as a scale drawing.
Instead: Use the numeric sides and angles; the sketch is normalised.
More verified examples
For the AAS inputs a=5, A=30° and B=60°, the missing angle is 90°; the SAS example a=3, b=4, C=60° gives area 5.1962 square units.
Both examples are generated from the same tested calculation used by the form. The source list and equations describe the conventions used by the solver.
Sources and methodology
The solve rules follow standard Euclidean trigonometry; each displayed example is calculated by the tested triangle logic.
The law of sines and law of cosines describe side-angle relationships. Heron’s formula gives area from three sides; triangle area identities link area with radii and semi-perimeter. Sources were accessed 22 October 2026. Page details show the byline, review date and changelog.
Enter exactly three known values: three sides (SSS), two sides and their included angle (SAS), two angles and one side (ASA or AAS), or two sides and a non-included angle (SSA). At least one value must be a side to set the scale.
Why can SSA have two answers?
The sine rule can give an angle and its supplement. Both may leave a positive third angle, so a=6, b=8, A=30° produces two valid triangles. The calculator shows both rather than choosing one.
Can three angles determine a triangle?
No. The angles fix the shape but not its size. Enter at least one side as well as two angles.
What units should I use?
Use the same length unit for all three side values. Angles must be in degrees. The tool labels lengths as units and area as square units because you choose the length unit.
Why are there two area values?
The result compares Heron's formula with half of ab sin C. They are equivalent for a valid triangle; for the 3–4–5 default both give 6 square units.
What do heights and medians mean?
A height is the perpendicular distance from a vertex to the opposite side. A median joins a vertex to the midpoint of the opposite side. Each is reported for all three sides.
Is the sketch drawn to scale?
No. The labelled SVG is normalised to fit its box, so it shows orientation and side/angle labels rather than exact proportions.
How does the calculator handle nearly right or equal-sided triangles?
It classifies side equality within a relative tolerance of 1e−9 times the largest side, and right angles within 1e−9 degrees. These tolerances affect labels only, not measurements.
Sources and methodology
The formulas are standard time-value-of-money results. We check every figure against golden values worked out independently, by closed form and by month-by-month simulation; see the methodology.
Law of Sines, Wolfram MathWorld (accessed 22 Oct 2026)
Law of Cosines, Wolfram MathWorld (accessed 22 Oct 2026)
Triangle Area, Wolfram MathWorld (accessed 22 Oct 2026)
Written and maintained by Nitin Raturi, a software engineer (about the author). Last reviewed 22 Oct 2026; next review due by . Found an error? Tell us.
Changelog
: First release. The 3–4–5 example has area 6 and perimeter 12; the 5–30°–60° AAS example has angle C=90°.