Triangles are the simplest polygons, but they come in several important types. Classifying triangles helps you describe their shape quickly and choose the right geometry tools. A triangle can be classified by the lengths of its sides, by the measures of its angles, or by both at the same time.
These categories are useful in proofs, constructions, measurement, and real-world design.
Side classification compares whether a triangle has no equal sides, two equal sides, or three equal sides. Angle classification compares whether all angles are less than 90 degrees, one angle is exactly 90 degrees, or one angle is greater than 90 degrees. Since every triangle has both side lengths and angle measures, combined names such as isosceles right triangle or scalene obtuse triangle give a more complete description.
The angle sum rule, a + b + c = 180 degrees, is the main tool for checking and finding triangle angle classifications.
Key Facts
- Every triangle has exactly 3 sides, 3 vertices, and 3 interior angles.
- Triangle angle sum: a + b + c = 180 degrees.
- Scalene triangle: all three side lengths are different.
- Isosceles triangle: at least two side lengths are equal.
- Equilateral triangle: all three side lengths are equal, and each angle is 60 degrees.
- Angle types: acute has all angles less than 90 degrees, right has one 90 degree angle, and obtuse has one angle greater than 90 degrees.
Vocabulary
- Scalene triangle
- A triangle with all three side lengths different.
- Isosceles triangle
- A triangle with at least two equal side lengths and two equal base angles.
- Equilateral triangle
- A triangle with three equal side lengths and three 60 degree angles.
- Right triangle
- A triangle with one interior angle that measures exactly 90 degrees.
- Obtuse triangle
- A triangle with one interior angle greater than 90 degrees.
Common Mistakes to Avoid
- Calling any triangle with two equal sides equilateral is wrong because equilateral requires all three sides to be equal.
- Forgetting that an equilateral triangle is also isosceles is wrong because it has at least two equal sides.
- Classifying a triangle only by its sides when the question asks for a complete classification is incomplete because triangles can also be classified by angles.
- Using angle measures that do not add to 180 degrees is wrong because every triangle must satisfy a + b + c = 180 degrees.
Practice Questions
- 1 A triangle has side lengths 7 cm, 7 cm, and 10 cm. Classify it by its sides.
- 2 A triangle has angle measures 35 degrees, 55 degrees, and 90 degrees. Classify it by its angles, and state whether the angle sum is valid.
- 3 A triangle has two equal sides and one angle greater than 90 degrees. Explain the most specific combined classification and why it cannot be equilateral.