Triangles: Classification, Special Triangles and Angle Sum
A triangle is a polygon with three sides and three angles, and it can be classified by its sides (scalene, isosceles, equilateral) or by its angles (acute, right, obtuse); the interior angles always add up to 180°. In brief: if two angles of a triangle are 50° and 65°, the third is 180° − 50° − 65° = 65°, so the triangle is also isosceles. The sum of any two sides is always greater than the third side.
Triangles are the simplest polygons and the building blocks of all others, because every polygon can be divided into triangles. Their rigidity makes them important in engineering. This guide explains the two classifications, the angle sum and the exterior angle theorem, the triangle inequality and the properties of special triangles, with a table of types.
How are triangles classified by sides?
- Scalene: all three sides have different lengths, and so do the angles.
- Isosceles: at least two sides are equal. The angles opposite to the equal sides (base angles) are equal. The line from the apex to the middle of the base is an axis of symmetry.
- Equilateral: all three sides are equal, and all three angles are 60°. It has three axes of symmetry.
In a triangle the larger side lies opposite the larger angle. For a triangle with sides 5, 6 and 8, the largest angle is opposite the side 8.
How are triangles classified by angles?
- Acute: all three angles are less than 90°.
- Right: one angle equals 90°. The longest side, the hypotenuse, is opposite it.
- Obtuse: one angle is greater than 90°.
A triangle can have at most one right or obtuse angle. The two classifications can be combined: an isosceles right triangle has angles 45°, 45° and 90°, and an equilateral triangle is always acute.
Why do the angles add up to 180°?
Through one vertex, draw a line parallel to the opposite side. The alternate angles formed by the parallel lines are equal to the two other angles of the triangle, and together with the third angle they make a straight angle of 180°.
Example 1: finding an angle
A triangle has angles 2x, 3x and 4x. Then 9x = 180°, so x = 20°, and the angles are 40°, 60° and 80°. All are less than 90°, so the triangle is acute and scalene.
Example 2: an isosceles triangle
The apex angle is 40°. The base angles are equal: (180° − 40°)/2 = 70° each. If instead one base angle is 40°, then the other base angle is also 40° and the apex is 100°, so the triangle is obtuse.
What is the exterior angle theorem?
An exterior angle is formed by one side of the triangle and the extension of an adjacent side. Its measure equals the sum of the two non-adjacent interior angles. If a triangle has angles 50° and 60° at A and B, the exterior angle at C is 110°, which is 50° + 60°, and also 180° − 70° (since the interior angle at C is 70°). The exterior angle is larger than either of these remote interior angles.
The three exterior angles, one at each vertex, always add up to 360°.
What is the triangle inequality?
Three lengths a, b, c can form a triangle if and only if each is less than the sum of the other two: a < b + c, b < a + c and c < a + b. It is enough to check that the longest is less than the sum of the other two.
Example 3
Sides 3, 4, 8: 3 + 4 = 7 < 8, so there is no triangle. Sides 5, 7, 11: 5 + 7 = 12 > 11, so a triangle exists. If the longest side equals the sum of the others, the three points are collinear and the triangle is degenerate. Given two sides 6 and 9, the third side x must satisfy 3 < x < 15.
Classification table
| Type | Defining property | Example angles | Example sides |
|---|---|---|---|
| Equilateral | 3 equal sides | 60°, 60°, 60° | 4, 4, 4 |
| Isosceles acute | 2 equal sides, all angles < 90° | 70°, 70°, 40° | 5, 5, about 3.4 |
| Isosceles right | 2 equal sides, one 90° | 45°, 45°, 90° | 1, 1, √2 |
| Isosceles obtuse | 2 equal sides, one angle > 90° | 30°, 30°, 120° | 1, 1, √3 |
| Scalene acute | no equal sides, all angles < 90° | about 47°, 58°, 75° | 6, 7, 8 |
| Scalene right | no equal sides, one 90° | 30°, 60°, 90° | 1, √3, 2 |
| Scalene obtuse | no equal sides, one angle > 90° | about 29°, 46°, 105° | 2, 3, 4 |
The sides of a 30°-60°-90° triangle are in the ratio 1 : √3 : 2. A 3-4-5 triangle is also a scalene right triangle, but its acute angles are about 36.9° and 53.1°. In the isosceles obtuse row, the side √3 follows from the law of cosines for sides 1 and 1 with an included angle of 120°.
Where do these facts come from?
The angle sum theorem is in Euclid’s Elements (Book I, Proposition 32). The statement that the base angles of an isosceles triangle are equal is also in the Elements and was nicknamed the “bridge of fools” in the Middle Ages, because it was the first difficult proposition for students. In non-Euclidean geometry, the angle sum is not 180°: on a sphere it is larger and in hyperbolic geometry it is smaller.
Where are triangles used today?
Bridges, roof trusses and towers use triangles, because a triangle cannot be deformed without changing the length of a side. Surveyors measure distances by triangulation. Computer graphics models surfaces as meshes of triangles. Maps and GPS positioning rely on the geometry of triangles.
Which mistakes are common?
Trusting how a figure looks
A triangle that looks isosceles may not be. Use the given marks and measurements rather than appearance.
Forgetting the triangle inequality
Not all triples of numbers are the sides of a triangle. Check before using formulas.
Mixing up isosceles and equilateral
Every equilateral triangle is isosceles, but not conversely.
Giving an exterior angle the wrong value
The exterior angle equals the sum of the two remote interior angles, not 180° minus their sum.