Classifying Triangles

Written by

Malcolm McKinsey

January 20, 2023

Fact-checked by

Paul Mazzola

What is a triangle?

A triangle is the simplest polygon. Its identifying property is its three sides enclosing a space, forming three interior angles. Because the definition is so simple, triangles can fall into categories based on their sides and their angles.

Polygons, like people, cannot often be described as all one thing or all another. You, for example, may like peanut butter but also enjoy playing chess. The two are unrelated, so we cannot leap to a conclusion that "only people who play chess like peanut butter." Triangles, in a similar way, can be described in many ways. Sometimes one triangle can match several descriptions.

Classifying triangles by sides

When you focus only on the lengths of triangles' sides, you get three classifications:

• Scalene - No two sides of equal length

• Isosceles - Two sides of equal length

• Equilateral - Three sides of equal length

Right away, you might sense trouble. All equilateral triangles (three equal-length, or congruent, sides) are also isosceles (two congruent sides).

Classifying triangles by angles

When you focus on the interior angles of triangles, you can get other classifications:

• Right - One right angle (90°) and two acute angles

• Oblique - No right angles

Oblique triangles can be further classified in one of two ways:

• Acute - All interior angles are acute, or each less than 90°

• Obtuse - One interior angle is obtuse, or greater than 90°

So we have seven ways to describe and classify triangles.

Multiple classifications

Can those seven descriptions ever overlap? Sure, because you could have a right triangle that is also scalene. You have to be careful how you mix-and-match your descriptions, because some things cannot overlap:

• An acute triangle cannot also be obtuse or right

• A scalene triangle cannot also be equilateral

Generally you want to evaluate the sides and then the angles, so you could get something like an equilateral triangle that is also acute.

Symbols to know

We use symbols in geometry. The symbol for a triangle is a tiny triangle: $\triangle$.

Here are other symbols you should recognize:

• Angle: $\angle$

• Congruent: $\cong$

• Degree: $°$

• Right angle: $□$

Classifying Triangles Examples

Enough talk; let's see these triangles! Draw or imagine △ABC, which is named for its interior angles.

The little tick marks on sides AB, BC and CA indicate they are equal in length to each other (congruent), so you know this is an equilateral triangle. Since all triangles have interior angles adding to 180°, we know:

We also know those three angles have to be congruent, because their opposite sides are congruent. Take the total 180° and divide it among the three congruent angles, and each angle must be 60°.

So our equilateral triangle is also an acute triangle, which means it is also an oblique triangle.

Here is a tricky one: is it also isosceles? Yes, because two of its sides are congruent. Some mathematics teachers, in order to simplify geometry, specify that an isosceles triangle has exactly two equal sides. Most mathematicians think more broadly, including equilateral triangles as isosceles.

Classifying Triangles: Example #2

Usually you can choose better letters than just A−B−C for the vertices of a triangle. Picture a very tall, very narrow triangle, △HUT. The two legs are 4 m tall, and base UT 2 m long.

It would not be a very comfortable hut, but it is a great △HUT, with a short base and two tall sides with tick marks showing they are congruent.

The two sides are measured as 4 m, and the base is measured as 2 m. Are all sides congruent? No. Are two sides congruent? Yes. So it is an isosceles triangle. You can also see that its three interior angles are less than 90°, so it is also acute and oblique.

Is that right?

Here we have △SAD with sides 6, 8, and 10 cm long, with a little square tucked into the corner of ∠A.

That square means ∠A is 90°. Since the interior angles add to 180°, the other two angles must be acute. This is a right triangle, having one right angle.

Now look at the sides. The sides are marked with their lengths to make your job easier: Side SA = 6 cm, AD = 8 cm, and the hypotenuse, SD = 10 cm long. Are the three lengths equal to one another?

No, so this is a right scalene triangle.

I saw some fleas!

A handy way to remember the word "isosceles" is the rhyme, "I saw some fleas," which you might see on the legs of a dog. The two equal-length sides of an isosceles triangle are called legs.

Imagine a △DOG, but ∠D has that little square in it, indicating ∠D is a right angle.

Check the sides first. Side DO is congruent to side GD, so our triangle is isosceles.

Now check the angles. Because we know ∠D is a right angle, we know the base angles are congruent and together add to 90°, which means they are both 45°. We have an isosceles right triangle!

Big finish

Let's imagine one last triangle. It is very wide and not very tall. It is almost the opposite of △HUT. We will call it △BIG because, well, it's big!

△BIG with horizontal side BG at top, obtuse ∠I at bottom, and side BI = 4 yds, side IG = 5 yds, side BG = 7.7 yds, and ∠G at 28.4° but verify graphically/visually.

First notice the side measurements:

• BI = 4yds

• IG = 5yds

• BG = 7.7yds

We have three sides of unequal length; △BIG is scalene.

Now move on to the interior angles. Check out ∠I, which is measured as 116°. That means ∠B and ∠G have to add up to only 64°, making them both acute. That makes one obtuse angle and two acute angles, so not only is △BIG scalene, it is also obtuse and oblique.

Lesson summary

Now that you made your way through the lesson, video and drawings, you can now identify triangles by the lengths of their sides and their interior angles, classify triangles according to their properties, and identify the same triangle using multiple descriptions and properties.