Enter the three side lengths. Angles, area, height, and perimeter are computed from them.
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What does this calculator estimate?
The triangle calculator solves a triangle from its three side lengths, returning the area, all three angles, heights, perimeter, inradius, and circumradius.
The triangle inequality
Three lengths form a triangle only if the sum of any two is greater than the third. If the largest side equals or exceeds the sum of the other two, the figure collapses into a line or cannot exist. This calculator checks that condition and warns when the sides are invalid.
Area from sides: Heron's formula
Heron's formula gives a triangle's area from its three sides alone: A = √(s(s−a)(s−b)(s−c)), where s is the semiperimeter. It is especially useful when you know all three sides but not the height.
Angles with the law of cosines
The law of cosines relates the sides to an angle: c² = a² + b² − 2ab·cos C. Rearranged, it gives each angle from the three sides. The three angles always sum to 180°, which this calculator uses as a consistency check.
How this calculator works
Formula
Area uses Heron's formula: A = √(s(s−a)(s−b)(s−c)), where s is the semiperimeter (a+b+c)/2. Angles come from the law of cosines, e.g. cos A = (b²+c²−a²)/(2bc). Height to side a is h = 2A/a; inradius r = A/s; circumradius R = abc/(4A).
Worked example
For a triangle with sides 5, 6, and 7, the semiperimeter is 9, the area is √(9···) = √(9×4×3×2) ≈ 14.70, and the angle opposite side 5 is about 44.4°.
Assumptions to verify
- You enter three side lengths that satisfy the triangle inequality (each side is shorter than the sum of the other two).
- Angles are reported in degrees.
- The triangle is assumed to be planar (Euclidean).
Frequently asked questions
How do you find the area of a triangle from its sides?
Use Heron's formula: A = √(s(s−a)(s−b)(s−c)), where s is half the perimeter. For sides 5, 6, and 7 the area is about 14.70 square units.
How do you find the angles of a triangle?
Use the law of cosines on each angle, then the fact that all three angles sum to 180°. This calculator computes all three from the three side lengths.
What is the triangle inequality?
A set of three side lengths forms a triangle only if each side is shorter than the sum of the other two. If not, no such triangle exists.
What is the inradius of a triangle?
The inradius is the radius of the largest circle that fits inside the triangle, tangent to all three sides. It equals the area divided by the semiperimeter (r = A/s).
What is the circumradius?
The circumradius is the radius of the circle that passes through all three vertices. It equals abc/(4A), where a, b, c are the sides and A is the area.
Cite this tool
BoringToolsKit. “Triangle Calculator.” boringtoolskit.com/triangle-calculator/ (reviewed August 2026). Free to reference in articles, syllabi, and answer posts with a link.
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