Area of Equilateral Triangle

What is an Equilateral Triangle?

Before we understand what an equilateral triangle is, let's first see the definition of a triangle. A triangle is a closed, 2-dimensional (2D) shape having 3 sides, 3 angles, and 3 vertices. Let us now define what an equilateral triangle is.
An equilateral triangle is a triangle in which all three sides are equal and the angles are also equal.

Understanding how to find its area transposes itself to be one very good exercise in math and further opens a lot of practical problems. In the following sections, we'll explore the properties and the formula for determining the area of an equilateral triangle.

Properties of Equilateral Triangle

Equilateral triangles possess unique features that distinguish them from other types of triangles. Below are a few important characteristics of equilateral triangles:

1. All three sides are of equal length.
2. All three angles have equal measure and are equal to 60^o.
3. Since all the sides and angles are equal, it is a regular polygon with 3 sides.
4. It exhibits rotational as well as line symmetry.
5. If we draw a perpendicular from the vertex of an equilateral triangle to the opposite side, it divides the triangle into two halves. Also, the angle of the vertex from where the perpendicular is drawn is divided into two equal angles, i.e., 30^oeach.
6. The triangle's median, altitude and angle bisector are identical for all sides.

Area of an Equilateral Triangle

The area of an equilateral triangle can be calculated using the formula:

Where,

→ A represents the area of the equilateral triangle.
→ s represents the length of the side of the equilateral triangle.

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Derivation of Area of an Equilateral Triangle

To derive the formula for the area of the equilateral triangle, we will use the general formula for the area of the triangle.
Now, consider the equilateral triangle ABC.
Draw a line AD perpendicular to the side BC, as shown in the given figure. It divides the triangle into two congruent right triangles.
Since the altitude bisects the base of the triangle, each side of one of the right triangles formed will be half the length of the equilateral triangle's side, i.e., BD = DC = $\frac{\frac{\mathrm{BC}}{}}{\frac{2}{}}$

Now that we have the altitude, we can use it to find the equilateral triangle's area by putting the altitude value in the general formula of the area of the triangle.

Thus, we have derived the formula for the area of an equilateral triangle.

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

Solved Examples of the Area of an Equilateral Triangle

Example 1: Calculate the area of an equilateral triangle with a side length of 10 cm.

Solution: Given: Length of side (s) = 10 cm
We know that the area of an equilateral triangle (A) = $\frac{\sqrt{3}}{4}$ × s²
Substituting the value of s in the formula:
A = $\frac{\sqrt{3}}{4}$ × 10²
= $\frac{\sqrt{3}}{4}$ × 100
= √3 × $\frac{\sqrt{100}}{4}$
= 100$\frac{\sqrt{3}}{4}$
= 25√3 cm² (approximately 43.3 cm² if we put the value of √3 = 1.732 )
Therefore, the area of an equilateral triangle is 43.93 cm².

Example 2: What is the area of an equilateral triangle with a side of 4 cm?

Solution:Given: Length of side (s) = 2 cm
We know that the area of an equilateral triangle = $\frac{\sqrt{3}}{4}$ × s²
Putting the value of s in the formula, we will get
Area = $\frac{\sqrt{3}}{4}$ × (4)²
= $\frac{\sqrt{3}}{4}$ × 16
= 4 √3 cm² (approximately 6.93 cm2. if we put  √3 = 1.732)
Therefore, the area of an equilateral triangle is 6.93 cm².

Frequently Asked Questions

1. What is the definition of an equilateral triangle?

Answer: An equilateral triangle is a type of triangle in which all three sides are of equal length.

2. Where do the orthocenter and centroid lie in a triangle? What is unique about them in an equilateral triangle?

Answer: In an equilateral triangle, the orthocentre and centroid both lie inside the triangle. In an equilateral triangle, the centroid, orthocenter, circumcenter and incenter all lie at the same point.

3. What is the formula for the equilateral triangle's perimeter?

Answer: The perimeter of an equilateral triangle can be calculated using the following formula:
P = 3 x s = 3s
where ‘s’ denotes the length of the side of an equilateral triangle