Classifying Triangles Calculator

Classify triangles by their sides (equilateral, isosceles, scalene) and angles (acute, right, obtuse). Enter three side lengths to identify the triangle type.

Last updated: April 2026 | By Patchworkr Team

Triangle Side Lengths

Side a

Side b

Side c

Classification Results

Results will appear here...

Triangle Classification Types

By Sides

Equilateral

All three sides equal

Isosceles

Two sides equal

Scalene

All sides different

By Angles

Acute

All angles < 90°

Right

One angle = 90°

Obtuse

One angle > 90°

How to Classify Triangles

Triangles can be classified in two ways: by their sides and by their angles. Every triangle has one classification from each category.

Classification by Sides:

Equilateral: All three sides are equal (a = b = c). These are always acute.
Isosceles: Exactly two sides are equal. The angles opposite the equal sides are also equal.
Scalene: All three sides have different lengths. Most triangles are scalene.

Classification by Angles:

Acute: All three angles are less than 90°. The sum still equals 180°.
Right: One angle is exactly 90°. Follows the Pythagorean theorem: c² = a² + b².
Obtuse: One angle is greater than 90°. The longest side is opposite the obtuse angle.

Classification Methods

Step 1: Verify Triangle Validity with Inequality

Check that a + b > c, a + c > b, and b + c > a (where c is the longest side). Why: Invalid side combinations don't form triangles. If the sum of two sides ≤ the third, the sides collapse into a line or don't form a closed shape. This must be verified first.

Step 2: Classify by Sides (Equilateral vs Isosceles vs Scalene)

Compare the three side lengths. All equal = equilateral; exactly two equal = isosceles; all different = scalene. Why: Side classification is independent and always applies. It tells you about the triangle's symmetry: equilateral has 3-fold symmetry, isosceles has 1-fold, scalene has none. This determines geometric properties like angle equality and center positions.

Step 3: Sort Sides and Calculate Angle Test Value

Let c be the longest side. Calculate c² and compare it to a² + b² (using the other two sides). Why: This sets up the Pythagorean test. The relationship between c² and a² + b² determines all angle sizes without needing to calculate each angle individually. It's the most efficient decision test.

Step 4: Apply Pythagorean Test for Angle Classification

If c² = a² + b² → Right triangle (one 90° angle)

If c² < a² + b² → Acute triangle (all angles < 90°)

If c² > a² + b² → Obtuse triangle (one angle > 90°)

Why: This relationship is fundamental geometry. It comes from the law of cosines: when c² = a² + b², the middle term (−2ab·cos(C)) equals zero, making cos(C) = 0, so C = 90°. No need to compute angles separately.

Step 5: Combine Classifications and Calculate Properties

Every valid triangle gets two labels: one from sides (equilateral/isosceles/scalene) AND one from angles (acute/right/obtuse). Calculate area using Heron's formula and perimeter as the sum of sides. Why: Full classification provides complete geometric understanding. Real applications (engineering, construction, surveying) need both the structural symmetry (side type) and angle profile. Area and perimeter complete the picture for material calculations and boundary specifications.

Real-World Example

Construction Roof Truss

Given:

A roof truss has support beams measuring 5m, 7m, and 8m. What type of triangle is formed?

Classify:

All sides different → Scalene

8² = 64, 5² + 7² = 74

64 < 74 → Acute

Result:

The truss forms a Scalene Acute triangle

Frequently Asked Questions

How do you classify a triangle?

Compare sides (equilateral, isosceles, scalene) and angles (acute, right, obtuse). Every triangle has one classification from each category.

Can a triangle be both equilateral and right?

No. Equilateral triangles have all 60° angles, so they're always acute, never right or obtuse.

What is the most common triangle type?

Scalene acute triangles are most common since they have no special restrictions on side lengths or angles.

How do I know if a triangle is right without measuring angles?

Use the Pythagorean theorem: if c² = a² + b² (where c is the longest side), it's a right triangle.

Can a triangle be both isosceles and obtuse?

Yes! An isosceles triangle can be acute, right, or obtuse, depending on the size of its angles.

What makes a triangle obtuse?

If the square of the longest side is greater than the sum of squares of the other two sides: c² > a² + b².

Are all scalene triangles acute?

No. Scalene means all sides are different—it can be acute, right, or obtuse. The 3-4-5 triangle is scalene and right.

What's the triangle inequality?

The sum of any two sides must be greater than the third side. Without this, the sides can't form a triangle.