Classifying Triangles Calculator — By Sides and Angles
Enter three side lengths to instantly classify the triangle by its sides (equilateral, isosceles, or scalene) and by its angles (acute, right, or obtuse), and see all three angles and the area.
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Perimeter
5 + 7 + 9 = 21 - 2
Semi-perimeter
21 ÷ 2 = 10.5 - 3
Heron's product s(s−a)(s−b)(s−c)
10.5 × 5.5 × 3.5 × 1.5 = 303.1875 - 4
Area = √(product)
√(303.1875) = 17.4123Square root of s(s−a)(s−b)(s−c) — matches the Area stat above.
How does this calculator work?
Enter sides a, b, c. By sides: equilateral (all equal), isosceles (two equal), scalene (all different). By angles: compute cos C = (a²+b²−c²)/(2ab) for the longest side c — right if C=90°, obtuse if C>90°, acute if all angles <90°. Area = √(s(s−a)(s−b)(s−c)) where s=(a+b+c)/2.
Formula
How this is calculated
Triangle classification has two dimensions: sides and angles. By sides, a triangle is equilateral if all three sides are equal (all angles are then 60°), isosceles if exactly two sides are equal, or scalene if all three sides differ. These are determined by comparing the entered lengths.
By angles, the calculator applies the law of cosines to the longest side c: cos C = (a² + b² − c²) / (2ab). If C = 90° the triangle is right (it satisfies the Pythagorean theorem a² + b² = c²). If C > 90° the triangle is obtuse; if all three angles are less than 90° it is acute. The two shortest sides always determine the largest angle, so only one angle can ever be right or obtuse.
All three angles are computed from the law of cosines and must sum to 180°. The area uses Heron's formula: Area = √(s(s−a)(s−b)(s−c)), where s = (a+b+c)/2 is the semi-perimeter. Inputs must be positive and satisfy the triangle inequality: each side must be strictly less than the sum of the other two.
Frequently asked questions
Yes. A 45-45-90 triangle has two equal legs and a right angle, making it isosceles and right. Similarly, a 45-45-90 triangle has legs of equal length. The two classifications (by side and by angle) are independent, so all six combinations except "equilateral right" and "equilateral obtuse" are theoretically possible.
For a valid triangle, each side must be strictly less than the sum of the other two (the triangle inequality). If side c ≥ a + b, the three sides cannot close into a triangle — they form a flat line or don't meet at all. Increase one of the shorter sides or reduce the longest.
A right triangle satisfies a² + b² = c² where c is the hypotenuse. The law of cosines generalises this: c² = a² + b² − 2ab·cos C. When C = 90°, cos C = 0 and the formula reduces exactly to the Pythagorean theorem.
Also known as
TG we-Calculate Editorial Team. (2026). Classifying Triangles Calculator — By Sides and Angles [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/classifying-triangles-calculator
TG we-Calculate Editorial Team. "Classifying Triangles Calculator — By Sides and Angles." TG we-Calculate. 2026. https://we-calculate.com/calculator/classifying-triangles-calculator.
TG we-Calculate Editorial Team, "Classifying Triangles Calculator — By Sides and Angles," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/classifying-triangles-calculator
@misc{wecalculate_classifying_triangles_calculator, title = {Classifying Triangles Calculator — By Sides and Angles}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/classifying-triangles-calculator}}, year = {2026}, note = {TG we-Calculate} }
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