Classifying Triangles Calculator
Enter three side lengths to instantly classify the triangle by sides and angles, and find all interior angles.
🔷 What is Classifying Triangles?
Classifying triangles means identifying the type of triangle based on its side lengths and interior angles. Every triangle belongs to exactly one side-based category and exactly one angle-based category, giving a combined classification such as "scalene right" or "isosceles obtuse". These labels carry precise geometric meaning and immediately reveal the triangle's symmetry, angle structure, and proportions.
Classification by sides has three categories. An equilateral triangle has all three sides equal in length; as a consequence, all three angles are exactly 60° and the shape is perfectly symmetric. An isosceles triangle has exactly two sides equal (the "legs"); the two angles opposite the equal sides (the "base angles") are always equal to each other. A scalene triangle has all three sides of different lengths, meaning all three angles are different as well. No two sides or angles are the same.
Classification by angles also has three categories. An acute triangle has all three interior angles strictly less than 90°. A right triangle has exactly one angle that equals 90° - the side opposite this right angle is the hypotenuse. An obtuse triangle has exactly one angle greater than 90°. A triangle can never have two obtuse angles or two right angles, because any two angles above 90° would already sum to more than 180°, leaving no room for the third.
This calculator determines both classifications automatically from three side lengths. It applies the Law of Cosines to compute all three interior angles, checks each classification rule, and reports the result along with the area (via Heron's formula), perimeter, and a plain-language description of the specific triangle type.
Triangle classification is a core topic in middle-school and high-school geometry, but it also has practical applications in architecture, structural engineering, and trigonometry. Knowing whether a triangle is right allows you to use the Pythagorean theorem directly. Knowing it is equilateral tells you all three angles are 60° without any calculation. This tool automates the classification and angle-finding steps so you can focus on understanding the geometry or verifying manual work.
Formulas Used
Step 1 - Law of Cosines (find all angles from three sides):
Repeat for angle B (swap a and b). Then C = 180° − A − B, from the angle sum property of triangles.
Step 2 - Classify by sides:
Step 3 - Classify by angles (using the largest side c):
Step 4 - Area via Heron's formula:
The perimeter is simply P = a + b + c, the sum of all three side lengths.
How to Use This Calculator
Steps to Classify Your Triangle
Example Calculations
Example 1 — 3-4-5 Right Scalene Triangle
Sides a = 3, b = 4, c = 5
Example 2 — 5-5-5 Equilateral Acute Triangle
Sides a = 5, b = 5, c = 5
Example 3 — 5-5-8 Isosceles Obtuse Triangle
Sides a = 5, b = 5, c = 8
Example 4 — 5-7-9 Scalene Obtuse Triangle
Sides a = 5, b = 7, c = 9
Example 5 — 5-5-7.071 Isosceles Right (45-45-90) Triangle
Sides a = 5, b = 5, c = 7.071 (approximately 5√2)
Frequently Asked Questions
🔗 Related Calculators
What are the types of triangles by sides?
Equilateral: all three sides equal (all angles 60°). Isosceles: exactly two sides equal (two base angles equal). Scalene: no sides equal (no angles equal). Every triangle falls into exactly one category.
What are the types of triangles by angles?
Acute: all three angles less than 90°. Right: exactly one angle equals 90°. Obtuse: exactly one angle greater than 90°. A triangle cannot have more than one right or obtuse angle.
Can a triangle be both isosceles and right?
Yes - the 45-45-90 triangle is both isosceles (two equal legs) and right (one 90° angle). Its sides are in ratio 1:1:√2. The two base angles are each 45°.
Can a triangle be both isosceles and obtuse?
Yes - for example, sides 5-5-8. The two equal sides are 5, and the base is 8. The apex angle (opposite the base) is obtuse. Using the Law of Cosines: cos(C) = (25+25-64)/(2×5×5) = -14/50 = -0.28, so C ≈ 106.3°.
How do you classify a triangle from its side lengths?
First check for equilateral (a=b=c), isosceles (any two equal), or scalene (all different). Then find the largest side c. If a²+b²>c², all angles<90° → acute. If a²+b²=c² → right. If a²+b²<c² → obtuse.
What is a scalene triangle?
A scalene triangle has all three sides different lengths and all three angles different. Most real-world triangles are scalene. There are no lines of symmetry. The 3-4-5 right triangle is scalene.
What determines if a triangle is acute?
A triangle is acute if all three angles are less than 90°. Equivalently, a triangle is acute if and only if: a²+b²>c², a²+c²>b², and b²+c²>a² - i.e. the square of each side is less than the sum of squares of the other two.
What is the largest possible angle in a triangle?
Just under 180°. In theory, as one angle approaches 180°, the triangle becomes flatter (degenerate). In practice, obtuse triangles have one angle between 90° and 180°. A triangle with exactly one 180° angle would be a straight line, not a triangle.