Table of Contents
Types of Triangles and How We Classify Them
In this chapter we look at how triangles are grouped into types. We will classify triangles in two independent ways:
- by their sides
- by their angles
Any particular triangle belongs to exactly one “side” type and exactly one “angle” type at the same time (for example, a triangle can be both isosceles and acute).
We assume you already know what a triangle is from the parent chapter.
Classification by Sides
Here we compare the lengths of the three sides.
Let a triangle have sides of lengths $a$, $b$, and $c$.
Scalene Triangle
A triangle is scalene if all three sides have different lengths.
That means:
$$
a \neq b,\quad b \neq c,\quad a \neq c
$$
No sides are equal, and usually all three angles are different as well.
Typical sketch: a triangle that looks “irregular,” with no obviously matching sides.
Isosceles Triangle
A triangle is isosceles if it has at least two sides of equal length.
Usually we say:
- $a = b$ (two equal sides) and $c$ is different.
In many courses, “isosceles” means exactly two equal sides, but in a strict mathematical sense, a triangle with all three sides equal (an equilateral triangle) also has “at least two equal sides,” so it fits the isosceles definition too. Whether an equilateral triangle is treated as a special kind of isosceles triangle depends on convention; most basic geometry courses still list it separately.
Important feature (to be proved elsewhere):
- In an isosceles triangle, the angles opposite the equal sides are equal.
We do not prove that here, but it is a key property used often.
Equilateral Triangle
A triangle is equilateral if all three sides have the same length:
$$
a = b = c
$$
Because all sides are equal, some important facts follow (used in other chapters):
- All three interior angles are equal.
- Since the sum of angles in a triangle is $180^\circ$, each angle must be $60^\circ$.
An equilateral triangle is highly symmetric and often used in examples and constructions.
Summary: By Sides
- Scalene: no equal sides.
- Isosceles: at least two equal sides (often taught as “exactly two equal”).
- Equilateral: all three sides equal.
Every triangle fits at least one of these descriptions.
Classification by Angles
Here we compare the sizes of the three interior angles.
Let the interior angles be $A$, $B$, and $C$ (measured in degrees or radians). Recall that in any triangle:
$$
A + B + C = 180^\circ
$$
Acute Triangle
A triangle is acute if all three interior angles are acute angles, that is, each is less than $90^\circ$:
$$
A < 90^\circ,\quad B < 90^\circ,\quad C < 90^\circ
$$
There is no right angle or obtuse angle in an acute triangle.
Right Triangle
A triangle is a right triangle if it has one right angle, i.e. an angle of $90^\circ$.
So, for some angle, say $C$,
$$
C = 90^\circ
$$
and the other two angles must add up to $90^\circ$.
Notationally:
- The side opposite the right angle is called the hypotenuse.
- The two sides that form the right angle are often called the legs.
Right triangles are especially important; many theorems (including the Pythagorean theorem) are specifically about them.
A triangle cannot have more than one right angle, because the sum of all angles must be $180^\circ$.
Obtuse Triangle
A triangle is obtuse if it has one obtuse angle, i.e. an angle greater than $90^\circ$:
For some angle, say $A$,
$$
A > 90^\circ
$$
and the other two angles must total less than $90^\circ$.
A triangle cannot have more than one obtuse angle, or the sum of angles would exceed $180^\circ$.
Summary: By Angles
- Acute triangle: all three angles are less than $90^\circ$.
- Right triangle: one angle is exactly $90^\circ$.
- Obtuse triangle: one angle is greater than $90^\circ$.
Every triangle fits exactly one of these three angle types.
Combining the Classifications
A single triangle has:
- one type based on side lengths, and
- one type based on angle sizes.
These can be combined. For example:
- Acute scalene triangle: all angles acute, all sides different.
- Acute isosceles triangle: all angles acute, two equal sides.
- Right isosceles triangle: one right angle, two equal sides (often has angles $45^\circ, 45^\circ, 90^\circ$).
- Obtuse scalene triangle: one obtuse angle, all sides different.
An equilateral triangle is always acute, because all three angles are $60^\circ$:
- Acute equilateral triangle: the only possible kind of equilateral triangle.
There is no such thing as:
- a “right equilateral triangle” (it would require a $90^\circ$ angle, but all angles are $60^\circ$), or
- an “obtuse equilateral triangle,” for the same reason.
Using Side Lengths to Detect the Angle Type
Later, when you study the Pythagorean theorem, you will see a useful connection between side lengths and angle classification.
Let a triangle have side lengths $a$, $b$, and $c$, with $c$ the longest side. Under the triangle inequality (so these can form a triangle), you have:
- If $a^2 + b^2 = c^2$, the triangle is right.
- If $a^2 + b^2 > c^2$, the triangle is acute.
- If $a^2 + b^2 < c^2$, the triangle is obtuse.
This provides a numerical way to classify triangles by angles using just their side lengths. The justification belongs in the Pythagorean theorem–related material, so here we only state the result.
Why Classification Matters
Classifying triangles is not just naming shapes; it helps you:
- know which theorems apply (for example, some results are only for right triangles, others for isosceles triangles),
- recognize special properties (like equal angles or sides) that simplify problems,
- organize geometric reasoning: when you see “isosceles” or “right” in a problem, you know immediately which tools might be useful.
In later chapters, many results will be stated with phrases like “In an isosceles triangle…” or “In a right triangle…”. This chapter gives the vocabulary needed to understand precisely what those phrases mean.