Classifying Triangles
Triangles are classified by sides (scalene, isosceles, equilateral) and angles (acute, right, obtuse). Enter three side lengths to determine the full classification with step-by-step reasoning.
Why This Mathematical Concept Matters
Why: Triangle classification is fundamental in geometry, 3D graphics, and surveying. The Euler line connects notable centers in non-equilateral triangles.
How: Use the triangle inequality to verify validity. Use the law of cosines to find angles. Compare sides for side classification; compare largest angle to 90° for angle classification.
- ●Scalene: all sides different. Isosceles: exactly two equal. Equilateral: all three equal.
- ●Acute: all angles < 90°. Right: one = 90°. Obtuse: one > 90°.
- ●The Euler line passes through centroid, circumcenter, and orthocenter.
Triangle Classification — Six Types, Two Systems
Identify scalene, isosceles, equilateral, acute, right, or obtuse triangles. Master the triangle inequality, Law of Cosines, and Euler line.
🔺 Common Triangle Examples — Click to Load
Triangle Sides
Triangle Visualization
Side Lengths Comparison
Angle Distribution
Side Proportions (Doughnut)
Triangle Properties Radar
Step-by-Step Breakdown
⚠️For educational and informational purposes only. Verify with a qualified professional.
🧮 Fascinating Math Facts
The Euler line passes through centroid, circumcenter, and orthocenter.
— Wolfram MathWorld
Equilateral triangles are the only type where all four notable centers coincide.
— Cut-the-Knot
Key Takeaways
- Triangles are classified by sides (scalene, isosceles, equilateral) and angles (acute, right, obtuse)
- Scalene: all sides different. Isosceles: exactly two sides equal. Equilateral: all three sides equal
- Acute: all angles < 90°. Right: one angle = 90°. Obtuse: one angle > 90°
- The triangle inequality theorem must hold: the sum of any two sides must exceed the third
- Every triangle has exactly one side classification and one angle classification — they combine (e.g., isosceles right)
Did You Know?
How Triangle Classification Works
Classification uses two independent systems. Every triangle has one type from each.
Classification by Sides
Scalene: a ≠ b ≠ c — all sides different. Isosceles: exactly two sides equal (a = b or b = c or a = c). Equilateral: a = b = c — all sides equal. Note: equilateral is a special case of isosceles. Use our Equilateral Triangle Calculator for specialized calculations.
Classification by Angles
Using the Law of Cosines, we compute all three angles. Acute: all angles < 90°. Right: one angle = 90° (Pythagorean theorem: a² + b² = c² when c is the longest side). Obtuse: one angle > 90°. See our Triangle Angle Calculator for angle computations.
Triangle Inequality Theorem
Before classifying, we verify the sides can form a triangle: a + b > c, a + c > b, b + c > a. If any inequality fails, the three lengths cannot form a valid triangle. This is essential in surveying and construction.
Expert Tips for Triangle Classification
Always Check Validity First
Verify the triangle inequality before classifying. Invalid side combinations produce meaningless angle classifications.
Equilateral Implies Acute
All equilateral triangles are acute (each angle is 60°). But not all acute triangles are equilateral.
Use Pythagorean Converse for Angles
Sort sides so a ≤ b ≤ c. Then a² + b² = c² ⇒ right; > c² ⇒ acute; < c² ⇒ obtuse. No need to compute angles for angle classification.
Euler Line and Triangle Centers
For non-equilateral triangles, the centroid, circumcenter, and orthocenter lie on a single line — the Euler line. This connects classification to deeper geometry.
Triangle Types at a Glance
| Type | By Sides | By Angles | Key Property |
|---|---|---|---|
| Scalene | All sides different | — | Most general |
| Isosceles | Exactly 2 sides equal | — | Symmetry axis |
| Equilateral | All 3 sides equal | Always acute (60° each) | All centers coincide |
| — | — | Acute | All angles < 90° |
| — | — | Right | One angle = 90° |
| — | — | Obtuse | One angle > 90° |
Frequently Asked Questions
Can a triangle be both isosceles and right?
Yes! An isosceles right triangle has two equal legs and one 90° angle. The two acute angles are each 45°. The sides follow the ratio 1 : 1 : √2.
Are all equilateral triangles acute?
Yes. Every equilateral triangle has three 60° angles, so all angles are less than 90°. Equilateral triangles are always acute.
How do I know if three numbers form a triangle?
Use the triangle inequality: a + b > c, a + c > b, and b + c > a. All three must hold. For example, 2, 3, 10 fails because 2 + 3 = 5 < 10.
Can an obtuse triangle be isosceles?
Yes. An isosceles obtuse triangle has two equal sides and one angle greater than 90°. The obtuse angle is opposite the unique side.
What is the most common triangle type?
Scalene triangles are most common — they have the fewest constraints. Random side lengths usually produce scalene triangles.
What is the Euler line?
The Euler line passes through the centroid, circumcenter, and orthocenter of any non-equilateral triangle. Leonhard Euler proved this in 1765.
What are triangle centers?
Key centers include: centroid (center of mass), circumcenter (center of circumscribed circle), incenter (center of inscribed circle), and orthocenter (where altitudes meet).
Why use Law of Cosines for angles?
When you know all three sides, the Law of Cosines directly gives each angle: cos(A) = (b² + c² − a²)/(2bc). It generalizes the Pythagorean theorem.
Triangle Classification by the Numbers
Official & Trusted Sources
Disclaimer: This calculator provides mathematically precise classifications based on standard geometric formulas. Results use floating-point arithmetic (~15 significant digits). For educational and general use. Not a substitute for professional surveying or engineering analysis.