About

Misclassifying a triangle leads to incorrect formula application. Using the Pythagorean theorem on a non-right triangle produces wrong results. Applying equilateral symmetry assumptions to a scalene shape causes structural miscalculations in engineering and architecture. This calculator accepts three side lengths a, b, c and applies the Triangle Inequality Theorem to validate feasibility. It then uses the Law of Cosines to derive all three interior angles and classifies the triangle by both sides (Equilateral, Isosceles, Scalene) and angles (Acute, Right, Obtuse). The tool assumes Euclidean plane geometry. Results degrade for degenerate triangles where the sum of two sides approaches the third.

Beyond classification, the calculator computes area via Heron's formula using semi-perimeter s = a + b + c2, along with inradius r, circumradius R, all three altitudes, and all three medians. Angle tolerance for right-angle detection uses ε = 0.0001 radians. Pro tip: floating-point arithmetic means a triangle with sides 1, 1, 1.4142135 will classify as right, not isosceles-right, unless you account for rounding in your inputs.

Formulas

Classification requires computing all three interior angles from three known sides. The Law of Cosines provides each angle.

A = arccos(b² + c² − a²2bc)

Where A is the angle opposite side a, and b, c are the adjacent sides. The same formula applies cyclically for angles B and C.

Area (Heron's): K = √s(s − a)(s − b)(s − c)

Where s = a + b + c2 is the semi-perimeter.

Inradius: r = Ks

Circumradius: R = abc4K

Altitude to side a: h_a = 2Ka

Median to side a: m_a = 12√2b² + 2c² − a²

Classification rules: By sides - if a = b = c, equilateral. If exactly two sides equal, isosceles. Otherwise, scalene. By angles - if any angle = 90° (within ε), right. If any angle > 90°, obtuse. Otherwise, acute.

Reference Data

Triangle Type	By Sides	By Angles	Example Sides	Interior Angles	Key Property
Equilateral Acute	Equilateral	Acute	5, 5, 5	60°, 60°, 60°	All sides and angles equal
Isosceles Acute	Isosceles	Acute	5, 5, 6	53.13°, 53.13°, 73.74°	Two equal sides, all angles < 90°
Isosceles Right	Isosceles	Right	1, 1, 1.414	45°, 45°, 90°	Two equal legs, hypotenuse = a√2
Isosceles Obtuse	Isosceles	Obtuse	3, 3, 5	33.56°, 33.56°, 112.89°	Two equal sides, one angle > 90°
Scalene Acute	Scalene	Acute	5, 6, 7	44.42°, 57.12°, 78.46°	No equal sides, all angles < 90°
Scalene Right (3-4-5)	Scalene	Right	3, 4, 5	36.87°, 53.13°, 90°	Pythagorean triple
Scalene Right (5-12-13)	Scalene	Right	5, 12, 13	22.62°, 67.38°, 90°	Pythagorean triple
Scalene Right (8-15-17)	Scalene	Right	8, 15, 17	28.07°, 61.93°, 90°	Pythagorean triple
Scalene Right (7-24-25)	Scalene	Right	7, 24, 25	16.26°, 73.74°, 90°	Pythagorean triple
Scalene Obtuse	Scalene	Obtuse	3, 4, 6	26.38°, 36.34°, 117.28°	No equal sides, one angle > 90°
30-60-90 Special	Scalene	Right	1, 1.732, 2	30°, 60°, 90°	Side ratios 1 : √3 : 2
Golden Gnomon	Isosceles	Obtuse	1, 1, 1.618	36°, 36°, 108°	Related to golden ratio φ
Near-Degenerate	Scalene	Obtuse	1, 2, 2.99	3.83°, 7.65°, 168.52°	Area approaches zero, nearly collinear

Frequently Asked Questions

Floating-point arithmetic means angles computed via arccos rarely equal exactly 90°. This calculator uses an epsilon tolerance of 0.0001 radians (approximately 0.006°). If the computed angle falls within 89.994° to 90.006°, it classifies as a right triangle. For critical engineering work, verify inputs are exact Pythagorean triples (e.g., 3-4-5, 5-12-13) to guarantee correct classification.

If any single side is greater than or equal to the sum of the other two (a ≥ b + c), the three segments cannot form a closed triangle. The calculator validates this condition before computation and returns an explicit error. A degenerate case (a = b + c) produces a triangle with zero area - effectively a line segment - and is rejected.

No. An equilateral triangle has all angles equal to 60°. A right triangle requires one angle of 90°. These conditions are mutually exclusive in Euclidean geometry. The only triangle that is both isosceles and right has angles 45°-45°-90°, with the hypotenuse equal to a leg multiplied by √2.

For a right triangle, the circumradius R equals exactly half the hypotenuse (R = c/2). For acute triangles, the circumcenter lies inside the triangle. For obtuse triangles, the circumcenter lies outside. The formula R = abc/(4K) holds universally, where K is the area. As a triangle approaches degeneracy (area → 0), the circumradius approaches infinity.

These are independent classification axes. A triangle can be isosceles-acute, isosceles-right, or isosceles-obtuse. Knowing only the side classification (e.g., 'isosceles') does not determine the angle classification. Both are needed to fully characterize the triangle's geometry. Engineering standards and textbook problems typically require the combined classification (e.g., 'scalene right triangle').

An interior angle can approach but never reach 180°. As one angle approaches 180°, the other two approach 0° and the triangle degenerates into a line. Practically, angles above 170° produce extremely flat triangles with near-zero area and very large circumradii. The calculator handles these edge cases but warns about numerical instability.