How to classify a triangle

A triangle can be classified in two ways: by the lengths of its sides and by the sizes of its interior angles. For example, a triangle with sides 3, 4 and 5 is both scalene and right.

Select an input method, enter three side lengths or three angles, and click Classify triangle. The calculator checks whether the inputs form a valid triangle, identifies both types, and shows a matching diagram.

Types of triangles by sides

Type	Rule	Example sides
Equilateral	All three sides are equal.	5, 5, 5
Isosceles	At least two sides are equal.	5, 5, 6
Scalene	All three sides are different.	3, 4, 5

Under the inclusive definition, an equilateral triangle is also isosceles. This calculator reports equilateral when all three sides are equal and uses isosceles when exactly two are equal.

Types of triangles by angles

Type	Rule	Example angles
Acute	All three angles are less than 90°.	60°, 60°, 60°
Right	One angle equals 90°.	30°, 60°, 90°
Obtuse	One angle is greater than 90°.	30°, 40°, 110°

Formulas for classifying triangles

1. Check the triangle inequality

All sides must be positive. After sorting them as x ≤ y ≤ z, a valid triangle satisfies:

x + y > z

If x + y = z, the triangle is flat. If x + y < z, the sides cannot meet to form a triangle.

2. Compare the squared side lengths

With z as the longest side:

x² + y² > z² → Acute triangle

x² + y² = z² → Right triangle

x² + y² < z² → Obtuse triangle

3. Calculate the interior angles

The cosine rule gives each angle from the three side lengths:

A = cos⁻¹[(b² + c² − a²) / (2bc)]

B = cos⁻¹[(a² + c² − b²) / (2ac)]

C = cos⁻¹[(a² + b² − c²) / (2ab)]

Angles are expressed in degrees. In a valid planar triangle, A + B + C = 180°. Equal angles have equal opposite sides, so angle inputs also identify the side classification.

4. Find perimeter and area from sides

Perimeter P = a + b + c

Semiperimeter s = (a + b + c) / 2

Area Δ = √[s(s − a)(s − b)(s − c)]

Perimeter uses the input length unit; area uses that unit squared. Angles alone do not determine perimeter or area because similar triangles can have different sizes.

Worked example: classify sides 3, 4 and 5

1. Check validity: 3 + 4 > 5, so the sides form a triangle.
2. Compare the sides: all three lengths differ, so it is scalene.
3. Compare squares: 3² + 4² = 9 + 16 = 25 = 5², so it is right.
4. Final answer: a scalene right triangle. Its angles are approximately 36.8699°, 53.1301° and 90°.

Its perimeter is 12 units and its area is 6 square units.

Example: classify angles 40°, 40° and 100°

The angles total 180°, so they form a valid triangle. Two equal angles mean two equal opposite sides. The 100° angle is greater than 90°. Therefore, this is an isosceles obtuse triangle.

Frequently asked questions

Can a triangle be both isosceles and right?

Yes. A 45°–45°–90° triangle has two equal sides and one right angle.

Is an equilateral triangle always acute?

Yes. Each interior angle of an equilateral triangle is 60°, which is less than 90°.

Can a scalene triangle be acute, right or obtuse?

Yes. Scalene describes unequal side lengths. Depending on its angles, a scalene triangle can be acute, right or obtuse.

Why are my side lengths invalid?

Each side must be positive, and the sum of the two shorter sides must exceed the longest side. For example, 2, 3 and 6 cannot form a triangle.

Can I classify a triangle using only its angles?

Yes. The largest angle determines whether it is acute, right or obtuse. Equal angles identify equal opposite sides. Angles alone cannot determine the triangle’s actual side lengths or area.

Numerical note: equality comparisons use a relative tolerance of 10⁻⁹ for sides and 10⁻⁷ degrees for angles. Angle totals within 10⁻⁷ degrees of 180° are normalized to 180°. Extremely flat triangles are rejected when reliable calculation is not possible.