Classifying Triangles Calculator

Classify any triangle by its sides (equilateral, isosceles, scalene) and by its angles (acute, right, obtuse) with detailed step-by-step analysis.

Enter Triangle Measurements

Classification

Triangle Type
Isosceles Acute
triangle
By SidesIsosceles
By AnglesAcute
Angle A---
Angle B---
Angle C---
Area---
Perimeter---

Step-by-Step Classification

Classification by sides and angles

How to Classify Triangles

Triangles can be classified in two ways: by the lengths of their sides and by the measures of their angles. Understanding these classifications is a foundational concept in geometry that helps in identifying properties, solving problems, and applying the correct formulas.

Classification by Sides

Equilateral Triangle

All three sides are equal in length. All three angles are also equal (60 degrees each).

a = b = c

Isosceles Triangle

Exactly two sides are equal in length. The angles opposite the equal sides are also equal.

a = b (or a = c or b = c)

Scalene Triangle

All three sides have different lengths. All three angles are also different.

a != b != c != a

Classification by Angles

Acute Triangle

All three angles are less than 90 degrees. Equivalently, a^2 + b^2 > c^2 for the longest side c.

All angles < 90 degrees

Right Triangle

One angle is exactly 90 degrees. Satisfies the Pythagorean theorem: a^2 + b^2 = c^2.

One angle = 90 degrees

Obtuse Triangle

One angle is greater than 90 degrees. Equivalently, a^2 + b^2 < c^2 for the longest side c.

One angle > 90 degrees

Using Side Lengths to Determine Angle Type

You can classify a triangle's angle type using only its side lengths without actually computing the angles. Let c be the longest side. Then:

  • If a2 + b2 > c2, the triangle is acute.
  • If a2 + b2 = c2, the triangle is right.
  • If a2 + b2 < c2, the triangle is obtuse.

Combined Classifications

Every triangle has a two-part classification: one for sides and one for angles. This gives six possible combinations: equilateral acute (the only option for equilateral), isosceles acute, isosceles right, isosceles obtuse, scalene acute, scalene right, and scalene obtuse. Note that equilateral triangles are always acute.

The Triangle Inequality

For three side lengths to form a valid triangle, they must satisfy the triangle inequality theorem: the sum of any two sides must be greater than the third side. This means a + b > c, a + c > b, and b + c > a must all hold true.

The Angle Sum Property

The interior angles of any triangle always sum to exactly 180 degrees. This fundamental property allows you to find a missing angle if the other two are known, and is essential for verifying that three given angles can form a valid triangle.

Finding Angles from Sides

Using the Law of Cosines, you can find any angle from the three side lengths:

  • Angle A = arccos((b2 + c2 - a2) / (2bc))
  • Angle B = arccos((a2 + c2 - b2) / (2ac))
  • Angle C = 180 - A - B