Types of Triangles — Definition, Formula & Examples
Types of triangles are the different ways to sort triangles into groups based on their sides or their angles. By sides, triangles can be scalene, isosceles, or equilateral; by angles, they can be acute, right, or obtuse.
Triangles are classified along two independent axes. By side length: a scalene triangle has no congruent sides, an isosceles triangle has at least two congruent sides, and an equilateral triangle has three congruent sides. By interior angle measure: an acute triangle has three angles less than 90°, a right triangle has exactly one 90° angle, and an obtuse triangle has exactly one angle greater than 90°. Every triangle belongs to exactly one category from each axis.
How It Works
To classify a triangle, check two things separately. First, compare its three side lengths — are any of them equal? That tells you whether it is scalene, isosceles, or equilateral. Second, look at its largest angle — is it less than 90°, exactly 90°, or greater than 90°? That tells you whether it is acute, right, or obtuse. A single triangle always gets two labels, one from each group. For example, a triangle with sides 3, 3, and 4 and all angles under 90° is an "acute isosceles triangle."
Worked Example
Problem: A triangle has sides of length 5 cm, 12 cm, and 13 cm. Classify it by sides and by angles.
Step 1 — Check the sides: Compare the three side lengths: 5, 12, and 13. No two sides are equal, so the triangle is scalene.
Step 2 — Check for a right angle: Test whether the sides satisfy the Pythagorean relationship. Square each side and see if the two smaller squares add up to the largest square.
Step 3 — Classify by angle: Because the equation holds, the triangle contains a 90° angle. It is a right triangle.
Answer: The triangle is a right scalene triangle.
Another Example
Problem: A triangle has angles of 60°, 60°, and 60°. Classify it by sides and by angles.
Step 1 — Check the angles: All three angles are 60°, which is less than 90°. The triangle is acute.
Step 2 — Check the sides: When all three angles are equal, all three sides must also be equal. The triangle is equilateral (and also isosceles, since it has at least two equal sides).
Answer: The triangle is an acute equilateral triangle.
Visualization
Why It Matters
Classifying triangles is one of the first geometry skills taught in elementary and middle-school math, and it builds the foundation for later topics like congruence and area formulas. Architects and engineers identify triangle types to choose the right structural support — for instance, right triangles appear in ramps and roof trusses. Knowing the type also tells you which formulas and theorems apply, saving time on homework and tests.
Common Mistakes
Mistake: Thinking that an equilateral triangle is not isosceles.
Correction: Isosceles means at least two equal sides. An equilateral triangle has three equal sides, so it is a special case of isosceles.
Mistake: Trying to classify by sides and angles using only one label.
Correction: Every triangle has two classifications — one by sides and one by angles. Always give both, such as "obtuse scalene" or "acute isosceles."