# Equilateral Triangles

There are many objects around us that are in the form of triangles like pizza slices, tower tops and roofs, and birthday banners. But sometimes we come across triangular shapes which look exactly the same in any direction we rotate it, like a nacho chip or traffic signs. Are these some kind of special form of triangles? Are they really equal in all ways? Let's find out.

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In geometry, triangles can be classified into different forms based on their sides and angles. And one of these forms is an equilateral triangle. In this section, we will understand the concept of the equilateral triangle and see its properties and formulas based on it.

A triangle is an equilateral triangle if it has three congruent sides. In other words, if all three sides of a triangle are of the same length, then it is an equilateral triangle.

So the name equilateral is derived from equi, which means equal, and lateral, means sides.

Equilateral triangle with congruent sides, Mouli Javia - StudySmarter Originals

## Equilateral triangles and angles

We can classify equilateral triangles based on their angles too.

An equilateral triangle is a triangle with all three of its internal angles congruent and equal to$60°$ .

Equilateral Triangle with same angles, Mouli Javia - StudySmarter Originals

### Corollaries on equilateral triangles

Let's take a look at some important statements regarding equilateral triangles

#### Corollary 1

Statement: Each angle of an equilateral triangle is$60°$.

Proof: To prove this consider$∆XYZ$an equilateral triangle.

$\therefore XY=YZ=ZX$

Now, an equilateral triangle is also considered an isosceles triangle. So we can apply the properties of the isosceles triangle to the equilateral triangle. Here we use Isosceles Triangle Theorem.

For that take:

$XY=YZ$ and $YZ=ZX$

$\therefore \angle Z=\angle X$ and $\angle X=\angle Y$

$⇒\angle X=\angle Y=\angle Z$

Now consider one of the properties of a triangle, which states that the sum of all internal angles of a triangle is equal to$180°$:$\angle X+\angle Y+\angle Z=180°$

As all three angles are equal, we can just consider one of them instead of all.

$\therefore \angle X+\angle X+\angle X=180°$

$⇒3\angle X=180°$

$⇒\angle X=\frac{180°}{3}$

$\therefore \angle X=60°$

So, $\angle X=\angle Y=\angle Z=60°$.

Hence, we can say an equilateral triangle is an equiangular triangle.

The Isosceles Triangle theorem states that angles opposite to the two sides of a triangle are equal if these two sides are equal.

From this corollary, we arrive at the next corollary.

#### Corollary 2

Statement: A triangle is equiangular if and only if it is equilateral.

## Equilateral Triangle Properties

Here are some of the properties of equilateral triangles:

1. An equilateral triangle is a regular polygon as it has three sides.

2. All sides and angles of equilateral triangles are congruent.

3. A perpendicular line drawn from any vertex of an equilateral triangle to its opposite side bisects both side and angle.

4. This perpendicular line (as mentioned above) is the same line for altitude, median, perpendicular bisector, and angle bisector for the same side.

5. Lines of symmetry in equilateral triangles are the three mentioned lines from each side.

6. In equilateral triangles, the centroid, ortho-center, circumcenter, and incenter are at the same point.

Remember that to bisect means to divide or split into two equal parts.

## Equilateral Triangle Formulas

Let's discuss a few formulas related to equilateral triangles, including its:

• Perimeter
• Area
• Height

### Perimeter of an Equilateral Triangle

Perimeter is the sum of all the sides. And as we are talking about an equilateral triangle, here all sides are equal. So the perimeter of an equilateral triangle is three times the length of one side.

Perimeter of an equilateral triangle$\mathbf{=}\mathbf{3}\mathbit{a}$. Here$a$ is the side length.

From this, we can deduce the formula for Semi Perimeter. Semi Perimeter is half of the perimeter of an equilateral triangle and we can calculate it as follows.

Semi Perimeter of an equilateral triangle$\mathbf{=}\mathbf{}\mathbf{3}\mathbf{a}}{\mathbf{2}}$

We usually use the semi perimeter to calculate the area of a triangle using Heron's formula.

What is the perimeter for the given equilateral triangle with a side of 6 cm? Also, find the semi perimeter for it.

Equilateral triangle, Mouli Javia - StudySmarter Originals

Solution: Here$a=6cm$. So applying the formula of perimeter, we get:

Perimeter of an equilateral triangleid="2552506" role="math" $=3a=3×6=18cm$.

Semi perimeter of an equilateral triangleid="2552508" role="math" alt="" $=3a}{2}=18}{2}=9cm$.

### Area of an Equilateral Triangle

Area is calculated to measure the space occupied within the sides of a polygon in a 2D plane. The formula to find the area of an equilateral triangle is as follows.

Area of an equilateral triangle$\mathbf{=}\mathbf{}\frac{\sqrt{\mathbf{3}}}{\mathbf{4}}{\mathbit{a}}^{\mathbf{2}}$, where$a$ is side length.

We can also calculate area using Heron's formula if a semi perimeter is given. Heron's formula is as follows.

Area of an equilateral triangle$\mathbf{=}\sqrt{\mathbf{s}{\mathbf{\left(}\mathbf{s}\mathbf{-}\mathbf{a}\mathbf{\right)}}^{\mathbf{3}}}$, where a is the side length and s is the semi perimeter of the triangle.

Calculate the area for an equilateral triangle with a side of 5 cm.

Equilateral triangle, Mouli Javia - StudySmarter Originals

Solution: Here$a=5cm.$

Area of an equilateral triangle $=\frac{\sqrt{3}}{4}{a}^{2}$

$=\frac{\sqrt{3}}{4}{\left(5\right)}^{2}$

$=10.83$

Therefore, the area of a given equilateral triangle is$10.83c{m}^{2}.$

### Height of an Equilateral Triangle

The height of an equilateral triangle is the perpendicular distance from a vertex of that triangle to its opposite side.

Height of equilateral triangle, Mouli Javia - StudySmarter Originals

The formula to calculate the height of an equilateral triangle is given below.

Height of an equilateral triangle$\mathbf{=}\mathbf{}\frac{\sqrt{\mathbf{3}}}{\mathbf{2}}\mathbit{a}$, where$a$ is the side length.

Find the height of an equilateral triangle with a side length of 15 cm.

Solution: Using the formula of height, we can say:

Height of an equilateral triangle$= \frac{\sqrt{3}}{2}a$

$= \frac{\sqrt{3}}{2}\left(14\right)$

$= 7\sqrt{3}=12.12cm$

Hence the height (or altitude) of an equilateral triangle is 12.12 cm.

## Examples of equilateral triangles

Now we work on some examples based on the above theory.

Find the area of an equilateral triangle that has a perimeter of 18 cm.

Solution: To find the area of an equilateral triangle, we need to know the length of its sides. So first we will find the side length using perimeter. We know that the formula for the perimeter of an equilateral triangle is$3a$. And the value of perimeter is also given in the question, which is 18 cm.

$\therefore 18=3a$

$⇒a=\frac{18}{3}⇒a=6cm$

Now as we have found the side length, we can use it in the formula of the area to calculate it.

Area of an equilateral triangle $=\frac{\sqrt{3}}{4}{\left(a\right)}^{2}$

$=\frac{\sqrt{3}}{4}{\left(6\right)}^{2}$

$=9\sqrt{3}=15.58c{m}^{2}$

Hence, an equilateral triangle that has a perimeter of 18 cm, has an area of 15.58 cm2.

An equilateral triangle with two side lengths is given. The length of one side is$\left(3x+8\right)$ and for the other side is$\left(4x+7\right)$. What is the measure of side length for this equilateral triangle? Also, find the perimeter for this triangle.

Solution: As the given triangle is an equilateral triangle, we know that all the sides of it are equal. So the given two side lengths are equal, and the equations can be set as equal to one another as well.

$⇒\left(3x+8\right)=\left(4x+7\right)$

To determine the side length we solve the above equation and find the value of x.

$⇒4x-3x=8-7⇒x=1$

Now, as both the side lengths are equal, we substitute the value of x in any one of the side lengths.

By substituting in$\left(3x+8\right)$, we get

$\left(3x+8\right)=\left(3\left(1\right)+8\right)= 11$.

We can check the correctness of the found value of x, by substituting x in both the side lengths. If both the value of side lengths are equal, the value of x is correct. Let's see for our case. We have already found the value of one of the side lengths. Let's find the other side length and compare it.

Substituting x in $\left(4x+7\right)$, we again get the value of 11. Hence as both the values of side length are equal, our calculated x value is correct!

Now that we have the sides' length, we can easily calculate the perimeter of the equilateral triangle.

Perimeter of an equilateral triangle$= 3a$. Here$a=11$.

$⇒3a=3\left(11\right)=33$.

So, the perimeter of the given equilateral triangle is 33 cm.

## Equilateral triangles - Key takeaways

• A triangle is an equilateral triangle if it has three congruent sides.
• An equilateral triangle is a triangle with all three of its internal angles congruent and equal to$60°$ .
• A triangle is equiangular if and only if it is equilateral.
• The perimeter of an equilateral triangle is$3a$.
• The semi perimeter of an equilateral triangle is$3\mathrm{a}}{2}$.
• The area of an equilateral triangle is$\frac{\sqrt{3}}{4}{a}^{2}$.
• The area of an equilateral triangle (using Heron's formula) is $\sqrt{s{\left(s-a\right)}^{3}}$.
• The height of an equilateral triangle is$\frac{\sqrt{3}}{2}a$.

#### Flashcards in Equilateral Triangles 8

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What is an equilateral triangle?

An equilateral triangle is a triangle with equal sides and angles.

How to find the area of an equilateral triangle?

The area of an equilateral triangle is calculated by the following formulas:

Area = (√3/4) × (side)2

How many lines of symmetry does an equilateral triangle have?

An equilateral triangle has three lines of symmetry.

What is the rule for equilateral triangles?

In an equilateral triangle, all three sides are congruent. Also, all angles are equal and have a measure of 60°.

What is the formula for equilateral triangles?

The formulas for an equilateral triangle are as follows:

Perimeter = 3 × side

Area =  (√3/4) × (side)2

Height = (√3/2) × side

## Test your knowledge with multiple choice flashcards

Are all angles in an equilateral triangle equal?

What is the perimeter of an equilateral triangle if the measure of one of its sides is 4cm?

Which of the following statement is correct?

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