Equilateral Triangle Formula

# Equilateral Triangle Formula

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## Introduction

A triangle having all three sides of equal length is known as an Equilateral triangle.

It can be represented by 3 similar marks on each side of the triangle. An equilateral triangle has all its interior angles equal to 60∘ .

## Equilateral Triangle Formulas

Side Length (s): In an equilateral triangle, all three sides have equal length. Let ‘s’ represent the length of a side.

Perimeter (P): The perimeter of an equilateral triangle is the sum of the lengths of all three sides. It can be calculated using the formula P = 3s.

Area (A): The area of an equilateral triangle can be determined using the formula

Where a is the length of the side of the triangle.

Height (h): The height of an equilateral triangle refers to the perpendicular distance from any vertex to the opposite side. It can be calculated as:

These formulas provide essential tools for calculating various properties of an equilateral triangle, such as its perimeter, area, and height.

## Solved Examples on Equilateral Triangle Formula

Example 1: Find the perimeter and area of an equilateral triangle with a side length of 5 cm. Solution:
Perimeter (P) = 3s
P = 3 x 5 cm
P = 15 cm
Area (A) = (sqrt(3) / 4) x s2
A = (sqrt(3) / 4) x (5 cm)2
A = (sqrt(3) / 4) x 25 cm2
A ≈ 10.83 cm2
Therefore, the perimeter of the equilateral triangle is 15 cm, and its area is approximately 10.83 cm2.

Example 2: Given the area of an equilateral triangle as 25 square units, find the side length. Solution:
Area (A) = (sqrt(3) / 4) x s2
25 = (sqrt(3) / 4) x s2
s2 = (25 x 4) / sqrt(3)
s2 ≈ 33.33
s ≈ sqrt(33.33)
s ≈ 5.77 units

Example 3: Find the perimeter of an equilateral triangle if its height is 12 cm.

Solution:

Height (h) = (sqrt(3) / 2) x s

12 cm = (sqrt(3) / 2) x s

s = (12 cm x 2) / sqrt(3)

s ≈ 13.86 cm

Perimeter (P) = 3s

P = 3 x 13.86 cm

P ≈ 41.59 cm

Example 4: In an equilateral triangle, the perimeter is 42 cm. Find the length of each side and the area.

Solution:

Let ‘s’ be the length of each side.

Perimeter (P) = 3s

42 cm = 3s

s = 14 cm

Area (A) = (sqrt(3) / 4) x s2

A = (sqrt(3) / 4) x (14 cm)2

A = (sqrt(3) / 4) x 196 cm2

A ≈ 98√3 cm2

## Frequently Asked Questions on Equilateral Triangle Formula:

1: What is the special property of equilateral triangle?

Answer: The special property of an equilateral triangle is that all three sides of the triangle are of equal length. This means that each angle in an equilateral triangle measures 60 degrees. In addition to having equal side lengths and equal angles, equilateral triangles also possess other interesting properties:

Equal Angles: Each angle in an equilateral triangle measures 60 degrees, making it an example of an acute triangle.

Symmetry: Equilateral triangles have multiple axes of symmetry. There are three lines of symmetry passing through the vertices, bisecting the opposite side and angle.

Regular Polygon: An equilateral triangle is a regular polygon, meaning all its sides and angles are equal. It is the simplest regular polygon, having only three sides.

2: Why is it called equilateral?

Answer: It is called equilateral because the term “equi-” means equal and “lateral” refers to the sides. An equilateral triangle has all three sides of equal length, distinguishing it from triangles with unequal sides.

3: What is the angle value of equilateral triangle?

Answer: The angle value of an equilateral triangle is 60 degrees for each of its three angles. In an equilateral triangle, all three angles are equal, and since the sum of angles in any triangle is always 180 degrees, each angle in an equilateral triangle measures 60 degrees. This property of equilateral triangles makes them regular polygons with equal angles and sides.

4: What is the formula for the area of equilateral triangle?

Answer: The formula for the area of an equilateral triangle is given by:

Where: A represents the area of the equilateral triangle, s represents the length of each side of the equilateral triangle.

5: Can an equilateral triangle have a right angle?

Answer: No, an equilateral triangle cannot have a right angle. Since all angles in an equilateral triangle are 60 degrees, there is no room for a 90-degree right angle.

6: Is an Equilateral Triangle a Regular Polygon Why?

Answer: An equilateral triangle is considered as a regular polygon or a regular triangle as its angles and sides are equal.

7: What is the Sum of all the Angles of an Equilateral Triangle?

Answer: The sum of all the angles of an equilateral triangle is equal to 180 degrees. As all the angles are equal to 60 degrees, the sum is equal to 60°+ 60°+ 60° = 180 degrees

8: What Are the Equilateral Triangle Formulas in Geometry?

Answer: In geometry, all the figures whether 2-D or a 3-D have certain specific formulas that are used to calculate the area, altitude, perimeter, and semi-perimeter. Similarly, an equilateral triangle has the below-listed formulas:

Area of an equilateral triangle, A = (√3/4)a2

Perimeter of an equilateral triangle, P = 3a

Semi perimeter of an equilateral triangle = 3a/2

Height of an equilateral triangle, h = (√3a/2)

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