Classifying Triangles: A Complete Guide to Triangle Types by Sides and Angles
Classify any triangle by its sides and angles with our free tool. Identify equilateral, isosceles, scalene, acute, right, or obtuse triangles instantly.
Triangles are classified along two independent axes: by their side lengths (equilateral, isosceles, or scalene) and by their angle measures (acute, right, or obtuse). This dual classification system produces up to nine distinct triangle types, each with unique properties and applications. From the perfectly symmetric equilateral triangle to the lopsided obtuse scalene, every triangle belongs to exactly one category on each axis. Understanding triangle classification is fundamental to geometry, trigonometry, and real-world applications ranging from structural engineering to computer graphics.
Key Takeaway
Triangles are classified by sides as equilateral (all equal), isosceles (two equal), or scalene (none equal), and by angles as acute (all < 90°), right (one = 90°), or obtuse (one > 90°). The Pythagorean comparison a² + b² vs c² determines the angle classification.
1. Classification by Sides
| Type | Definition | Example |
|---|---|---|
| Equilateral | All three sides equal | 5, 5, 5 |
| Isosceles | Exactly two sides equal | 5, 5, 8 |
| Scalene | No sides equal | 3, 4, 5 |
2. Classification by Angles
To classify a triangle by its angles, compare the sum of the squares of the two shorter sides to the square of the longest side:
If a² + b² = c² → Right triangle
If a² + b² > c² → Acute triangle
If a² + b² < c² → Obtuse triangle
| Type | Definition | Example |
|---|---|---|
| Acute | All angles < 90° | 3, 4, 5 (right) → no. 5, 5, 6 (acute) |
| Right | One angle = 90° | 3, 4, 5 |
| Obtuse | One angle > 90° | 2, 3, 4 |
3. Combined Classification
Combining both classifications gives nine possible types. Some combinations are impossible: an equilateral triangle must be acute (all angles are 60°), so "equilateral right" and "equilateral obtuse" triangles cannot exist.
4. The Triangle Inequality
Before classifying a triangle, verify that the given sides form a valid triangle. The triangle inequality states that the sum of any two sides must exceed the third: a + b > c, a + c > b, b + c > a. If any of these fail, the "triangle" is degenerate (flat) and cannot be classified.
5. Real-World Applications
Triangle classification is used in structural engineering (determining load distribution), computer graphics (mesh generation and rendering), surveying (land measurement), navigation (triangulation), and crystallography (classifying crystal structures).
6. Frequently Asked Questions
Can a triangle have two right angles?
No. The angles of a triangle must sum to 180°. Two right angles would sum to 180°, leaving no room for the third angle. A triangle can have at most one right angle.
Can an equilateral triangle be right?
No. An equilateral triangle has all angles equal to 60°, so it cannot have a 90° angle. Equilateral triangles are always acute.
How do I find the angles from three sides?
Use the law of cosines: cos A = (b² + c² − a²)/(2bc). Calculate each angle using inverse cosine. The largest angle is always opposite the longest side.
Using Our Calculator
Enter the three side lengths and the calculator instantly classifies the triangle by both sides and angles. It validates the triangle inequality, computes the area, and shows the complete classification with step-by-step explanations.