Classifying Triangles

A triangle is a polygon with three sides, three vertices, and three interior angles.

Although all triangles have these basic features, triangles can have very different shapes. Mathematicians classify triangles in two main ways:

• by the lengths of their sides
• by the sizes of their angles

An important idea is that a triangle can belong to one category from each system at the same time.

For example, a triangle could be both isosceles and right-angled.

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Classifying Triangles by Their Sides

Triangles can be classified by comparing the lengths of their three sides.

There are three main classifications:

• equilateral
• isosceles
• scalene

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Equilateral Triangles

An equilateral triangle has three equal sides.

Because all three sides are equal, all three interior angles are also equal.

Since the angles inside every triangle add to: 180^o

each angle in an equilateral triangle must be: 180^o ÷ 3 = 60^o

Therefore, an equilateral triangle always has:

• 3 equal sides
• 3 equal angles
• each angle equal to 60^o

• 

Equal sides are often shown on diagrams using matching tick marks.

If all three sides have the same tick mark, they have equal lengths.

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Isosceles Triangles

An isosceles triangle has at least two equal sides.

The angles opposite those equal sides are also equal.

These equal angles are often called the base angles.

For example, suppose an isosceles triangle has two equal angles of: 70^o

The third angle is: 180^o - 70^o - 70^o = 40^o

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Scalene Triangles

A scalene triangle has three sides of different lengths.

Its three angles are also different.

Therefore, a scalene triangle has:

• no equal sides
• no equal angles

For example, a triangle with sides: 5 cm, 7 cm, 9 cm

is scalene because all three side lengths are different.

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Comparing Side Classifications

Side markings on diagrams can help us identify the classification even when no measurements are provided.

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Classifying Triangles by Their Angles

Triangles can also be classified according to their angles.

The three main classifications are:

• acute
• right
• obtuse

The key angle to examine is usually the largest angle.

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Acute Triangles

An acute triangle has three angles that are all less than: 90^o

For example: 50^o 60^o 70^o

All three angles are less than 90^o

Therefore, the triangle is: Acute

An equilateral triangle is always acute because all three of its angles are 60^o

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Right Triangles

A right triangle contains exactly one angle equal to: 90^o

A right angle is usually shown using a small square symbol.

The side opposite the right angle is called the hypotenuse.

The hypotenuse is always the longest side of a right triangle.

The other two sides are often called the legs.

Right triangles are extremely important in mathematics because they are used in:

• the Pythagorean theorem
• trigonometry
• measurement
• construction
• navigation
• engineering

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Obtuse Triangles

An obtuse triangle contains one angle greater than: 90^o

but less than: 180^o

For example: 110^o 40^o 30^o

Because: 110^o > 90^o

the triangle is: Obtuse

A triangle cannot have two obtuse angles because their total would already exceed 180^o

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Comparing Angle Classifications

A useful memory rule is:

Acute → all angles < 90°

Right → one angle = 90°

Obtuse → one angle > 90°

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The Triangle Angle Sum

One of the most important properties of every triangle is:

$\boxed{\text{The interior angles add to }180^\circ}$

This property allows us to calculate missing angles.

If the three angles are A, B, and C: A + B + C = 180^o

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Finding a Missing Angle

Suppose a triangle contains angles of: 45^o 65^o x

Since the angles total 180^o

45 + 65 + x = 180

110 + x = 180

x = 70^o

The three angles are therefore: 45^o 65^o 70^o

All are less than 90^o

Therefore, the triangle is acute.

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A Triangle Can Have Two Classifications

Side classification and angle classification describe different properties.

Therefore, we can often give a triangle two names.

For example, consider a triangle with sides: 5 cm, 5 cm, 7 cm

and angles: 45^o 45^o 90^o

Two sides are equal.

Therefore:

Isosceles

One angle is 90^o

Therefore:

Right

The complete classification is:

Isosceles right triangle

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Possible Combinations

Several combinations are possible.

Acute Scalene

All sides are different and all angles are less than 90^o

Acute Isosceles

Two sides are equal and all angles are less than 90^o

Right Scalene

All sides are different and one angle is 90^o

Right Isosceles

Two sides are equal and one angle is 90^o

Obtuse Scalene

All sides are different and one angle is greater than 90^o

Obtuse Isosceles

Two sides are equal and one angle is greater than 90^o

Equilateral

An equilateral triangle is always acute, because: 60^o 60^o 60^o

So an equilateral triangle cannot be right or obtuse.

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Using Side Lengths to Classify Triangles

Suppose a triangle has sides: 8 cm, 8 cm, 5 cm

Two sides are equal.

Therefore: Isosceles

Now consider: 4 cm, 6 cm, 9 cm

All three sides are different.

Therefore: Scalene

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Using Angles to Classify Triangles

Suppose the angles are: 35^o 55^o 90^o

Because one angle equals 90^o

Right Triangle

Now consider: 25^o 45^o 110^o

Because one angle is greater than 90^o

Obtuse Triangle

Worked Example 1

An isosceles triangle has a vertex angle of: 40^o

Find the other two angles.

Because the triangle is isosceles, the other two angles are equal.

First find the remaining angle total:

180^o - 40^o = 140^o

Divide equally:

140^o ÷ 2 = 70^o

Therefore: 70^o 70^o

The triangle's angles are: 40^o 70^o 70^o

All angles are less than 90^o

So the complete classification is:

Acute isosceles triangle

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Worked Example 2

A triangle has angles:

$45^\circ,\quad45^\circ,\quad90^\circ$

Two angles are equal, so the sides opposite them are also equal.

Therefore, the triangle is isosceles.

It also contains a

$90^\circ$

angle.

Therefore:

$\boxed{\text{Isosceles right triangle}}$

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Worked Example 3

A triangle contains angles:

$32^\circ,\quad48^\circ,\quad x$

Calculate

$x$

:

$x=180^\circ-32^\circ-48^\circ$$x=100^\circ$

Because one angle is greater than

$90^\circ$

, the triangle is obtuse.

Since all three angles are different, all three opposite sides must also be different.

Therefore:

$\boxed{\text{Obtuse scalene triangle}}$

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The Longest Side and Largest Angle

There is an important relationship between the sides and angles of a triangle:

The longest side is opposite the largest angle.

Similarly:

The shortest side is opposite the smallest angle.

For example, if the angles are:

$40^\circ,\quad60^\circ,\quad80^\circ$

the side opposite

$80^\circ$

will be the longest.

The side opposite

$40^\circ$

will be the shortest.

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Can Any Three Lengths Make a Triangle?

No.

For three lengths to form a triangle, they must satisfy the triangle inequality.

The sum of the lengths of any two sides must be greater than the length of the third side.

Consider:

$3,\quad4,\quad5$

Check:

$3+4>5$

Yes.

These lengths can form a triangle.

Now consider:

$2,\quad3,\quad6$

Check:

$2+3>6$

This is false.

Therefore:

$\boxed{\text{These lengths cannot form a triangle.}}$

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Solving Triangle Problems

When solving a triangle-classification problem, use this strategy:

Step 1 – Examine the sides

Ask:

• Are all three equal?
• Are two equal?
• Are all different?

Classify as:

equilateral, isosceles, or scalene

Step 2 – Examine the angles

Ask:

• Are all below$90^\circ$?
• Is one exactly$90^\circ$?
• Is one greater than$90^\circ$?

Classify as:

acute, right, or obtuse

Step 3 – Use the angle sum if necessary

Remember:

$A+B+C=180^\circ$

Step 4 – Give both classifications when possible

For example:

$\boxed{\text{Obtuse isosceles triangle}}$

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Common Misconceptions

A triangle does not have to have only one classification.

A triangle can be both isosceles and right, because one term describes its sides and the other describes its angles.

A right triangle cannot contain two right angles.

Two

$90^\circ$

angles would already total

$180^\circ$

, leaving no angle for the third vertex.

An obtuse triangle cannot contain two obtuse angles.

Two angles greater than

$90^\circ$

would total more than

$180^\circ$

.

An equilateral triangle cannot be right or obtuse.

Its angles are always:

$60^\circ,\quad60^\circ,\quad60^\circ$

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Did You Know?

An equilateral triangle is also a special type of isosceles triangle under the mathematical definition that an isosceles triangle has at least two equal sides.

However, in many school classification exercises, the categories are treated separately:

• equilateral = three equal sides
• isosceles = two equal sides
• scalene = no equal sides

Always follow the definitions being used in your course or question.

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Key Terms

Triangle – A polygon with three sides and three angles.

Equilateral triangle – A triangle with three equal sides and three

$60^\circ$

angles.

Isosceles triangle – A triangle with at least two equal sides.

Scalene triangle – A triangle with no equal sides.

Acute triangle – A triangle with three angles less than

$90^\circ$

.

Right triangle – A triangle containing one

$90^\circ$

angle.

Obtuse triangle – A triangle containing one angle greater than

$90^\circ$

.

Hypotenuse – The longest side of a right triangle, opposite the right angle.

Triangle inequality – The rule that the sum of any two side lengths must be greater than the third side.

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Key Takeaways

• Triangles can be classified by their sides and angles.
• An equilateral triangle has three equal sides and three$60^\circ$angles.
• An isosceles triangle has at least two equal sides and corresponding equal angles.
• A scalene triangle has no equal sides.
• An acute triangle has three angles less than$90^\circ$.
• A right triangle has one angle equal to$90^\circ$.
• An obtuse triangle has one angle greater than$90^\circ$.
• The interior angles of every triangle add to$180^\circ$.
• A triangle can have both a side classification and an angle classification.
• The longest side is opposite the largest angle.
• The shortest side is opposite the smallest angle.
• The triangle inequality can be used to determine whether three lengths can actually form a triangle.
• Triangle properties can be used to calculate missing angles, compare sides, and solve geometric problems.