Triangle Calculator
Solve any triangle from 3 sides, 2 sides + angle, or 2 angles + side. Get all angles, area, perimeter, heights, and triangle type.
Select a mode and enter values.
How Does the Formula Work?
The triangle calculator solves any triangle when given enough information — three sides (SSS), two sides and the included angle (SAS), two angles and the included side (ASA), or two angles and a non-included side (AAS). It uses the Law of Cosines and Law of Sines to compute all missing values, then calculates area, perimeter, heights, and both the inscribed and circumscribed circle radii.
cos(A) = (b² + c² − a²) / (2bc)
Law of Sines (angles ↔ sides):
a/sin(A) = b/sin(B) = c/sin(C)
Area = ½ × a × b × sin(C)
Heron's: Area = √[s(s−a)(s−b)(s−c)] where s = (a+b+c)/2
Heights: hₐ = 2 × Area / a
Inradius: r = Area / s
Circumradius: R = abc / (4 × Area)
Example: a=3, b=4, c=5 (right triangle)
A=36.87°, B=53.13°, C=90°
Area=6, Perimeter=12, r=1, R=2.5
The Four Solving Modes
SSS (Side-Side-Side) is the most common case: given all three sides, the calculator uses the Law of Cosines to find each angle. SAS (Side-Angle-Side) uses the Law of Cosines to find the third side from two sides and their included angle, then calculates the remaining angles. ASA (Angle-Side-Angle) and AAS (Angle-Angle-Side) both use the fact that angles sum to 180° to find the third angle, then apply the Law of Sines to find the unknown sides. The calculator automatically validates that the input forms a valid triangle before computing.
Area Formulas
The calculator uses the trigonometric area formula (½ab sin C) which works for any triangle when two sides and the included angle are known. For SSS cases, this is equivalent to Heron's formula: Area = √[s(s−a)(s−b)(s−c)] where s is the semi-perimeter. Both formulas give identical results — the trigonometric form is computationally simpler since the angles are already calculated.
Triangle Classification
Every triangle has two classifications. By sides: equilateral (all three sides equal, all angles 60°), isosceles (exactly two sides equal, two angles equal), or scalene (all sides different). By angles: acute (all angles less than 90°), right (one angle exactly 90°), or obtuse (one angle greater than 90°). These six combinations create categories like "isosceles right triangle" (two equal sides with a 90° angle) or "scalene obtuse" (all sides different with one angle over 90°). The calculator detects and displays both classifications automatically.
Heights, Inradius, and Circumradius
The three heights (altitudes) connect each vertex to the opposite side perpendicularly, calculated as hₐ = 2 × Area / a. The inradius r is the radius of the incircle — the largest circle that fits inside the triangle, tangent to all three sides. The circumradius R is the radius of the circumscribed circle — the circle passing through all three vertices. For a right triangle with hypotenuse c, the circumradius equals c/2 (the circumscribed circle's diameter equals the hypotenuse). These geometric properties have applications in engineering, architecture, surveying, and navigation.
Tips & Recommendations
SSS, SAS, ASA, AAS — solve from sides, angles, or any combination. Dynamic form adapts to your chosen mode.
All 3 sides, all 3 angles, area, perimeter, 3 heights, inradius, and circumradius — everything in one click.
Equilateral, isosceles, or scalene. Right, acute, or obtuse. Both shape and angle type detected automatically.
Triangle inequality and angle sum checked. Invalid inputs show a clear error message — no wrong results.
Frequently Asked Questions
What solving modes are available?
SSS (3 sides), SAS (2 sides + included angle), ASA (2 angles + included side), and AAS (2 angles + non-included side). Each mode uses different formulas to solve the complete triangle.
How are the angles calculated from 3 sides?
Using the Law of Cosines: cos(A) = (b² + c² − a²) / (2bc). Each angle is calculated from the three known sides, and the result is converted from radians to degrees.
What is the inradius and circumradius?
The inradius is the radius of the largest circle that fits inside the triangle (incircle). The circumradius is the radius of the circle that passes through all three vertices (circumscribed circle).
How is the triangle type determined?
Shape is determined by side equality: equilateral (all equal), isosceles (two equal), scalene (all different). Angle type by largest angle: right (90°), acute (all < 90°), obtuse (one > 90°).
Why does my triangle show 'Invalid triangle'?
The triangle inequality must hold: the sum of any two sides must be greater than the third side. Also, the sum of any two angles must be less than 180°.
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