Equilateral Triangle

Triangles are one of the most fundamental figures of two-dimensional geometry in mathematics. A triangle has three sides and angles, and the sum of all the angles is equal to 180°. An equilateral triangle is a special case of these triangles with all sides and angles equal to each other. Due to their symmetrically perfect shape, equilateral triangles are the basic figures in both theoretical and practical geometry. So, let’s uncover the essential formulas and basic concepts for the equilateral triangle. 

1.0Equilateral Triangle Definition

An equilateral triangle can be defined as a triangle whose three sides are of equal length. An equilateral triangle not only has equal sides but also three equal angles measuring 60°. Therefore, the triangle is not only regular in the sense of its sides but regular in the sense of its angles as well. In simpler terms, an equilateral triangle is equiangular as well as equilateral.

Hence, by the above definition, one can easily visualise the shape of an equilateral triangle. For instance, an equilateral triangle shape can be characterised as:  

2.0Equilateral Triangle Properties 

The properties of an equilateral triangle are the fundamentals which make it different from other triangles. The unique properties of these triangles are: 

• Equal Sides and angles: This is the most well-known property of an equilateral triangle. The length of all three sides and angles is equal. If the length of each side is denoted by a, then all three sides are equal to a. Also, every interior angle of this triangle measures exactly 60°
• Symmetry: An equilateral triangle has three axes of symmetry, i.e., it can be divided into two halves in three different ways. The axes of symmetry are along the medians, i.e., the line segments from each vertex to the midpoint of the opposite side.
• Circumcenter, Centroid, and Incenter Coincide: In an equilateral triangle, the circumcenter (the centre of the circumscribed circle), centroid (the centre of mass), and incenter (the centre of the inscribed circle) all lie at the same point. This is due to the symmetry of the equilateral triangle. This point is equidistant from all three vertices and all three sides.
• Height (Altitude): The altitude (or height) of an equilateral triangle is the perpendicular from a vertex to the opposite side. It is especially crucial in determining the area and other features of the triangle.
• Equal Perpendicular Bisectors: The perpendicular bisectors of the sides of an equilateral triangle are equal in length and intersect at a common point, which is the circumcenter. This property also illustrates the symmetry of the figure.

3.0Equilateral Triangle Formulas

Now that we have understood the theoretical components of an equilateral triangle, let’s get an insight into some of its practical formulas: 

• Equilateral Triangle Perimeter: The equilateral triangle perimeter is the sum of all sides of the triangle. Since all the sides are equal, P can be mathematically written as: 

P=3 a

• Equilateral Triangle Area: The area of an equilateral triangle is the area enclosed by its edges. It can be calculated by using the formula: 

$A=\frac{\sqrt{3}}{4}{a}^2$

The formula mentioned here can be derived from the general formula for the area of a triangle itself. It includes the fact that the triangle has equal sides, which is responsible for the following formula. 

• Height (Altitude): The formula for the altitude of an equilateral triangle is based on the Pythagorean theorem, which can be mathematically written as: 

$h=\frac{\sqrt{3}}{2}\mathbf{a}$

It accounts for the fact that the altitude (perpendicular height) divides the equilateral triangle into two right triangles, each of which has one-half of the base and the entire height.

4.0Equilateral Triangle Theorem

Just like any other geometrical figure, equilateral triangles also possess some of the most important theorems. These theorems ultimately define the properties of an equilateral triangle, which include: 

Theorem 1: If a triangle is equilateral (all sides are equal), then it is also equiangular (all angles are equal to 60°).

To Prove: In an equilateral triangle, all angles are equal to 60°

Given: An equilateral triangle ABC with all sides equal to “a” and ∠A = ∠B = ∠C

Proof: The theorem can be proved using the triangle's angle sum property. It states that the sum of all the angles of any given triangle will always be equal to 180. Which means, 

∠A + ∠B + ∠C = 180°

Since ∠A = ∠B = ∠C, the equation can be rewritten as: 

∠A + ∠A + ∠A = 180°

3∠A = 180°

∠A = 60°

Theorem 2: If a triangle is equiangular (all angles are 60°), then it is equilateral (all sides are equal).

To Prove: A given triangle, say ABC, is an equilateral triangle. 

Given: In $△ABC$ , ∠A = ∠B = ∠C = 60°

Proof: The theorem can be proved with the converse of the isosceles triangle theorem. This theorem states that in any triangle, if two sides are equal, then the opposite angles are also equal. Therefore, we will prove that equal angles mean that the opposite sides are also equal. With the help of this theorem, we can say that: 

• Since ∠A = ∠B, AB = AC
• Similarly, since ∠B = ∠C, BC = AB.
• Lastly, because ∠C = ∠A, CA = BC.

Hence, the above conclusions show that: 

AB = AC = BC

Since all the sides of the triangle are equal, $△ABC$ is an equilateral triangle. 

5.0Equilateral Triangle Examples

Problem 1: The area of an equilateral triangle is given as $36\sqrt{3}{\mathrm{cm}}^2$.Find the side length of the triangle. 

Solution: Given that: 

$\begin{array}{c}A=\frac{\sqrt{3}}{4}{a}^2\\ 36\sqrt{3}=\frac{\sqrt{3}}{4}{a}^2\\ a=\sqrt{36\times 4}\\ a=6\times 2=12\mathrm{cm}\end{array}$

Problem 2: Find the height of an equilateral triangle having a side length of 12 cm.

Solution: Given that h = 12cm 

$\begin{array}{c}h=\frac{\sqrt{3}}{2}a\\ h=\frac{\sqrt{3}}{2}(12)=6\sqrt{3}\mathrm{cm}\end{array}$

Problem 3: A triangular garden in the shape of an equilateral triangle has a side length of 15 meters. The garden would be surrounded by a fence along all three sides. Additionally, a flower bed is to be planted in the center of the garden, and the area of the flower bed is $10\sqrt{3}{m}^2$.

1. Find the perimeter of the triangular garden.
2. Calculate the remaining area of the garden (the area not covered by the flower bed).

Solution: given that a = 15m

1. P=3 a=3(15)=45 m
2. Area of remaining garden = Area of the whole garden - Area of flower bed

$\text{ Area of remaining garden }=\frac{\sqrt{3}}{4}{a}^2-100\sqrt{3}\begin{array}{rl}& \text{ Area of remaining garden }=\\ & \frac{\sqrt{3}}{4}(15{)}^2-10\sqrt{3}=56.25\sqrt{3}-10\sqrt{3}=46.25\sqrt{3}{\mathrm{cm}}^2\end{array}$