Triangle Calculator
Fill in any three of the six values — at least one side — and the solver completes the triangle: sides, angles, area, perimeter, heights, medians, inradius, and circumradius. If two sides and a non-included angle fit two triangles, both are shown.
Method & second solution (if any)
How the solver picks its method
Three sides (SSS) go straight to the law of cosines. Two sides with the angle between them (SAS) use it once for the third side, then finish as SSS. One side with two angles (ASA/AAS) finds the third angle from the 180° sum and scales the sides with the law of sines. The tricky case is two sides with a non-included angle (SSA): the law of sines can return two valid angles — one acute, one obtuse — and when both survive the 180° check, two different triangles fit your numbers. The second is shown instead of discarded.
How it’s calculated
Law of cosines: a² = b² + c² − 2bc·cos A. Law of sines: a/sin A = b/sin B = c/sin C = 2R. Area by Heron’s formula: √(s(s−a)(s−b)(s−c)) with s the semiperimeter. Heights hₐ = 2·Area/a; medians mₐ = ½√(2b² + 2c² − a²); inradius = Area/s; circumradius = a/(2 sin A).
Results update as you type and are for education, not professional advice — double-check any number that matters.
Worked example
Sides a = 8, b = 6, c = 10: the angles come out A = 53.13°, B = 36.87°, C = 90° — a right triangle. Area = 24, perimeter = 24, heights 6 / 8 / 4.8, medians 7.2111 / 8.544 / 5, inradius 2, circumradius 5. An ambiguous case: a = 8, b = 10, A = 40° produces two triangles — B = 53.464°, C = 86.536°, c = 12.423 (area 39.93), or B = 126.536°, C = 13.464°, c = 2.898 (area 9.31).
Common mistakes
- Entering three angles — they fix the shape but not the size; at least one side is required.
- Treating the SSA case as unique — always check whether a second triangle appears below the results.
- Mixing units: an angle typed as 1.05 means 1.05° in degree mode, not radians — switch the unit first.
Where it is used
- Geometry and trigonometry homework with full working.
- Surveying, navigation, and roof/rafter layout from partial measurements.
- CAD sanity checks — verifying a sketch is actually constructible.
Frequently asked questions
Why do I sometimes get two answers?
With two sides and a non-included angle (SSA), the side opposite the given angle can swing to two positions — one acute, one obtuse — both consistent with your inputs. For a = 8, b = 10, A = 40° both B = 53.46° and B = 126.54° are valid, so two triangles exist. When only one survives the angle-sum check, only one is shown.
Why can’t three angles define a triangle?
Angles set only the shape. A 60-60-60 triangle can have any size — every equilateral triangle matches. You need at least one length to pin down scale.
What do the heights and medians mean?
A height is the perpendicular distance from a vertex to the opposite side (hₐ is drawn to side a); a median joins a vertex to the midpoint of the opposite side. The three medians meet at the centroid.
What are the inradius and circumradius?
The inradius is the radius of the largest circle that fits inside the triangle (Area ÷ s); the circumradius is the radius of the circle through all three vertices (a ÷ 2sin A). For the 6-8-10 example they are 2 and 5.