Triangles

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Mathematicians often use the known properties of different shapes to help them solve problems. In this article, we will explore the classic and common three-sided shape, the triangle. You may be surprised to see an article dedicated entirely to triangles, but this is a broad topic with many interesting details to uncover! Let's start by defining what we even mean by a triangle.

Meaning of triangles

The term "triangle" itself is a combination of two words: tri (meaning three) and angle (a space formed by the meeting of two lines). We can use this understanding to approach our definition of a triangle:

Triangles are shapes with three sides. Because they have three sides, they also have three angles.

Triangles used to be referred to as trigons. However, this term has mostly been replaced with the more common term, triangle.

Now, let's illustrate what we mean by a triangle. Every triangle has three sides and three edges or corners which are known as vertices.

The figure below shows a triangle, $A B C$ . We can write $△ A B C$ to denote the triangle $A B C$ . Now, $△ A B C$ has three vertices A, B, and C. It also has three sides: AB, BC, and CA.

Triangles Example of a triangle StudySmarter

Example of a triangle - StudySmarter Originals

Angles in triangles

As illustrated in the above image, triangles have three angles. If we were to cut each of these angles out of the triangle and line them up next to each other, we could notice that all three angles would form a straight line. Recall that angles on a straight line sum to 180 degrees. Therefore, we can say that angles in a triangle add up to 180 degrees.

Therefore, if the three angles of the triangle are $α$ , $β$ and $γ$ , we can say that:

$α + β + γ = 180 °$

This is an important fact, as we can use it to help determine missing angles in a triangle. We will do this in the following example:

Suppose we have a triangle with angles $30 °$ and $50 °$ . Work out the third angle.

Solution:

Let's denote the missing angle by $α$ . Since the three angles in a triangle add up to $180 °$ , we can say:

$30 ° + 50 ° + α = 180 °$

Therefore,

$80 ° + α = 180 °$ .

Subtracting $110 °$ from both sides, we obtain:

$α = 180 ° - 80 ° = 100 °$

Therefore, the missing angle is $100 °$ .

Area of triangles

Now, we will talk about finding the area of a triangle.

The area of a shape is the space that it takes up. It is measured in square units (i.e., m² or ft²).

There is a formula that allows us to work out the area of any given triangle. It is:

$A r e a o f a T r i a n g l e = \frac{1}{2} \times b a s e \times h e i g h t$

So, all we need to know is the base and the height to work out the triangle's area. When we refer to the height, we are talking about the perpendicular height as measured from the base. So, the height and base should be at right angles to each other, as shown in the diagram below.

area of a triangle- showing the perpendicular height of a triangle Triangle ACB with perpendicular height DC shown - StudySmarter Originals

In the triangle ACB, we have the base of $A B$ and the height of $C D$ . We can also see that $A B$ is perpendicular to $C D$ ( $A B ⊥ C D$ ). So, if we measured their lengths, we could work out the area of this triangle using the formula.

Recall that the area is measured in square units. So, if the height and the base are measured in centimeters ( $c m$ ), the area would be measured in centimeters squared ( $c m^{2}$ ).

Suppose the base of a triangle is $10 c m$ and the height is $12 c m$ . Work out the area of the triangle.

Solution:

Using the fact that:

$A r e a o f a T r i a n g l e = \frac{1}{2} \times b a s e \times h e i g h t$

We can say that:

$A r e a o f a T r i a n g l e = \frac{1}{2} \times 10 c m \times 12 c m = 60 c m^{2}$

Therefore, the area of this triangle is $60 c m^{2}$ .

The perimeter of triangles

In addition to the area of triangles, we are often asked to work out the perimeter as well. The perimeter is the sum of all of the lengths of the triangle's sides. So, to obtain the perimeter, we need to add up these side lengths.

The formula for a triangle's perimeter can be written as:

$P = a + b + c$

Where $a$ , $b$ , and $c$ are the lengths of each of the three sides of the triangle. Let's take a look at how to use this formula in an example problem.

If we have a triangle with side lengths $3 c m$ , $4 c m$ , and $5 c m$ , what would the perimeter be?

Solution:

Using the formula for the perimeter, we have that:

$P = 3 + 4 + 5 = 12 c m$

So, the perimeter of this triangle would be $12 c m$ .

Types of triangles

There are different types of triangles that are characterized by specific properties. We will discuss the properties of four types in more detail, including:

The equilateral triangle
The isosceles triangle
The scalene triangle
The right-angled triangle

Equilateral triangles

Equilateral triangles consist of three equal sides and three equal angles, which helps to explain the name of equilateral. Recall from earlier that the three angles in a triangle sum up to $180 °$ . Since the equilateral triangle has three equal angles, we can say that each angle is $60 °$ , as calculated by: $180 \div 3 = 60 °$ . If we have a triangle where we know each angle is equal to $60 °$ , we can say that it is an equilateral triangle.

The figure below shows an example of an equilateral triangle. Note that the ticks on each side of this triangle are there to show that each of the sides is equal in length.

Triangles image of equilateral triangle ABC StudySmarter Equilateral triangle ABC - StudySmarter Originals

Isosceles triangles

Isosceles is a fun word to say, but what does it mean? Isosceles triangles are triangles with two equal sides and hence two equal angles. So, a useful characteristic of isosceles triangles is that we only need to know the size of one of the angles to be able to work out the other two! We will look at an example of this later on.

Below is an example of an isosceles triangle. Note that the ticks on two of the sides show that these two sides are equal in length.

Triangles image of isosceles triangle DEF StudySmarter Example of an isosceles triangle - StudySmarter Originals

Scalene triangles

So, we know that an equilateral triangle has three equal sides, and an isosceles triangle has two equal sides. Can you guess what a scalene triangle is? Scalene triangles have no equal sides and no equal angles.

Below is an example of a scalene triangle. This time there are no ticks on any of the sides because none of the sides are the same!

Triangles image of scalene triangle GHI StudySmarter Example of a scalene triangle - StudySmarter Originals

Right-angled triangles

We also have a special type of triangle, which is instead classified by the properties of its angles. If one of the triangle's angles is a right angle, meaning it is $90 °$ , the triangle is a right-angled triangle. This type of triangle is particularly useful in the study of Trigonometry. Below is an example of a right-angled triangle:

Triangles example of right angled triangle StudySmarter Example of a right-angled triangle - StudySmarter Originals

Now, if we have a right-angled triangle, by definition, the triangle is also either an isosceles or scalene triangle. Take a look at the below example to see why:

Suppose the three angles of a triangle are $90 °$ , $30 °$ , and $60 °$ . In this case, since one of the angles is a right angle, it is a right-angled triangle. However, since all three of the angles are different, it is also a scalene triangle.

Now, suppose we have another right-angled triangle with angles of $90 °$ , $45 °$ , and $45 °$ . In this case, it is a right-angled triangle and also an isosceles triangle because two of the angles are the same.

It's not possible for a triangle to be both equilateral and right-angled, however. To fit the definition of an equilateral triangle, all of the angles would need to be the same, and to fit the definition of a right-angled triangle, one of the angles would need to be $90 °$ . This means that the triangle would need to have three angles of $90 °$ , like so:

$90 ° + 90 ° + 90 ° = 270 ° \neq 180$

However, the angles of a triangle have to add up to $180 °$ ! Thus, right-angled triangles can also be classified as either isosceles or scalene.

Pythagorean theorem

An important and well-known theorem about right-angled triangles is the Pythagorean theorem, which relates to the sides of right-angled triangles. This theorem is very useful because it enables us to find the length of a missing side of a right-angled triangle if we already know the other two sides.

Triangles Pythagorean theorem StudySmarter Right-angled triangle and Pythagorean theorem - StudySmarter Originals

For the right-angled triangle above, with sides labelled as $a$ , $b$ , and $c$ , the theorem gives the following formula:

$a^{2} + b^{2} = c^{2}$

The side labelled as $c$ is known as the hypotenuse of the triangle. Let's now take a look at a quick example to see how the Pythagorean theorem works.

Suppose we have the below triangle. Work out the size of the size labelled $x$ :

types of triangles- right angled triangle with missing side Right-angled triangle with missing side - StudySmarter Originals

Solution:

For this right-angled triangle, we can see that $x$ is the hypotenuse, so we label it as $c$ to fit our formula. So, let's now label the other sides as $a = 3$ and $b = 4$ .

Applying the Pythagorean theorem, we can say that:

$a^{2} + b^{2} = c^{2}$

Now, substituting in our values of $a$ , $b$ , and $c$ , we get:

$3^{2} + 4^{2} = x^{2}$

$9 + 16 = x^{2}$

$25 = x^{2}$

Taking the square root of both sides,

$x = \sqrt{25} = 5$

Therefore, the length of the triangle's hypotenuse is $x = 5 c m$ .

When we have integer values for all three sides of a right angle, the side lengths are together known as a Pythagorean Triple.

Triangle examples

We will now go through some example problems concerning triangles to test your understanding!

A triangle has two angles $52 °$ and $38 °$ . Show that this triangle is right-angled.

Solution:

Let's first define the missing angle to be $x °$ . Since angles in a triangle sum to $180 °$ , we have:

$52 ° + 38 ° + x ° = 180 °$

Therefore,

$90 ° + x ° = 180 °$

Subtracting $90 °$ from both sides, we obtain:

$x = 180 ° - 90 ° = 90 °$ .

Thus, the missing angle is $90 °$ , which is a right angle. From this, we know that it is a right-angled triangle.

In the below isosceles triangle $M N O$ , we know that $M N = O M$ and $∠ M N O = 42 °$ . Work out the size of the other two angles.

Triangles triangle example on finding missing angles StudySmarter Triangle example finding missing angle - StudySmarter Originals

Solution:

Since $M N = O M$ , we know that $∠ M O N = 42 °$ . Now, since angles in a triangle sum to $180 °$ , we can say:

$42 ° + 42 ° + ∠ N M O = 180 °$ .

Therefore,

$84 ° + ∠ N M O = 180 °$

Subtracting $84 °$ from both sides, we obtain:

$∠ N M O = 180 ° - 84 ° = 96 °$

So, $∠ M O N = 42 °$ and $∠ N M O = 96 °$

In the below triangle, $△ A D C$ is equilateral and $∠ C A B = 32 °$ . Work out the size of $∠ A C B$ and $∠ A B C$ .

Triangles triangle example on finding missing angles StudySmarter Triangle example finding missing angles - StudySmarter Originals

Solution:

Firstly, since $△ A D C$ is equilateral, we can say that each of the angles within it are $60 °$ . So, $∠ D C A = 60 °$ .

Since angles on a straight line sum to $180 °$ , we have:

id="2869227" role="math" $∠ A C B = 180 ° - ∠ D C A = 120 ° ∠ A C B = 180 ° - 60 ° = 120 °$

With this information, we can work out $∠ A B C$ :

id="2869236" role="math" $∠ A C B + ∠ C A B + ∠ A B C = 180 ° 120 ° + 32 ° + ∠ A B C = 180 °$

$152 ° + ∠ A B C = 180 °$

Subtracting $152 °$ from both sides, we get:

$∠ A B C = 180 ° - 152 ° = 28 °$ .

So $∠ A C B = 120 °$ and $∠ A B C = 28 °$ .

A given isosceles triangle has an angle of $30 °$ . Work out two possibilities for the sizes of its other two angles.

Solution:

Firstly, since it is isosceles, two of the angles must be the same. If one of the angles is $30 °$ , then one of the other angles could be $30 °$ as well to meet this property. In this case, that would make the third and last angle $120 °$ by the following calculation:

$180 ° - 30 ° - 30 ° = 120 °$

So, our isosceles triangle could have angles: $30 °, 30 °, 120 °$ .

Another possible scenario is that only one of the angles is $30 °$ . In this case, the other two angles would need to be the same. Since angles in a triangle sum to $180 °$ , the other two angles would need to sum to:

$180 ° - 30 ° = 150 °$ .

Since the two remaining angles are both the same, they would each be:

$150 ° \div 2 = 75 °$ .

Therefore, our isosceles triangle could also have angles: $30 °, 75 °, 75 °$ .

So, the two possibilities are: $30 °, 30 °, 120 °$ or $30 °, 75 °, 75 °$ .

Triangles - Key takeaways

Triangles are shapes with three sides and three angles.
Every triangle has three sides and three edges or corners which are known as vertices.
The three angles in a triangle add up to 180 degrees.
We have a formula for the area of a triangle as follows: $A r e a o f a T r i a n g l e = \frac{1}{2} \times b a s e \times h e i g h t$
The four main types of triangles are: equilateral, isosceles, scalene, and right-angled.
Equilateral triangles consist of three equal sides and three equal angles.
Isosceles triangles are triangles with two equal sides and two equal angles.
Scalene triangles have no equal sides and no equal angles.

Frequently Asked Questions about Triangles

A triangle is a shape with three sides.

The area of any triangle can be computed by multiplying 1/2 by the base, multiplied by height.

The internal angles in a triangle sum to 180 degrees.

To find the height of a triangle, determine the perpendicular distance from the base to the top vertex of the triangle.

To find the perimeter of a triangle, add up the lengths of all of the sides of the triangle.

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Flashcards in Triangles26 Start learning

What are congruence transformations?

Congruence transformations are transformations which when performed on an object produce congruent objects.

What are congruent triangles?

Congruent triangles are triangles with equal sides and angles.

What do you do when you are unable to determine if a pair of triangles are congruent?

In this case, you will have to prove that another pair of triangles are congruent and then use the information gotten about its congruent sides and angles to prove that the first pair is congruent.

What is a geometric use of congruent triangles?

Finding distances between points.

There is a story about one of Napoleon's generals using congruent triangles. What did he use them to determine?

He used them to determine the distance from one bank of a river to another.

How did Napoleon's general supposedly use congruent triangles?

The legend goes that the officer would stand at the bank of the river, and look down the brim of his cap until it was lined up with the opposite bank. Then, keeping his head at this level he would turn away from the river. Now, all he had to do was make a measurement from where he was standing to where the brim of his cap had lined up, and this would tell him the width of the river.

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