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.

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.

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.

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:

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.

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

$\Rightarrow 3\angle X=180°$

$\Rightarrow \angle X=\frac{180°}{3}$

$\therefore \angle X=60°$

So, $\angle X=\angle Y=\angle Z=60°$.

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

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.

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$\mathbf{=}\mathbf{3}\mathit{a}$. 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$\raisebox{1ex}{$\mathbf{=}\mathbf{}\mathbf{3}\mathbf{a}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{2}$}\right.$

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

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$\mathbf{=}\mathbf{}\frac{\sqrt{\mathbf{3}}}{\mathbf{4}}{\mathit{a}}^{\mathbf{2}}$, 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$\mathbf{=}\sqrt{\mathbf{s}{\mathbf{(}\mathbf{s}\mathbf{-}\mathbf{a}\mathbf{)}}^{\mathbf{3}}}$, where a is the side length and s is the semi perimeter of the triangle.

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$\mathbf{=}\mathbf{}\frac{\sqrt{\mathbf{3}}}{\mathbf{2}}\mathit{a}$, where$a$ is the side length.

Examples of equilateral triangles

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