**Equilateral Triangle Formula **

The Equilateral Triangle is another common type having all three sides are equal and the interior angles are also the same i.e. 60-degree.

The equilateral Triangle Formulas in geometry are given as –

\[ Area \;of \;an \;Equilateral \;Triangle = \frac{\sqrt{3}}{4}a^{2}\]

\[ Perimeter \;of \;an \;Equilateral \;Triangle = 3a\]

\[ Semi \;Perimeter \;of \;an \;Equilateral \;Triangle = \frac{3a}{2}\]

\[ Height \;of \;an \;Equilateral \;Triangle = \frac{\sqrt{3}}{2}a\]

Where,

a is the side of an equilateral triangle.

h is the altitude of an equilateral triangle.

Where a is the side length of an equilateral triangle and this is the same for all three sides.

#### Proof Area of Equilateral Triangle Formula

According to the properties of an equilateral triangle, the lengths of an equilateral triangle are the same for all three sides. we can write a = b = c

Area of triangle = \( \sqrt { s (s – b)(s – c)(s – a)} \)

=> we know that \( s =\frac{a+b+c}{2} \)

=>\( s =\frac{a+a+a}{2} = \frac{3a}{2} \)

Put the Value of S in the Heron’s Formulas

=> \( \sqrt { \frac{3a}{2} (\frac{3a}{2} – b)(\frac{3a}{2} – c)(\frac{3a}{2} – a)} \)

=> \( \sqrt { \frac{3a}{2} (\frac{3a-2b}{2})(\frac{3a-2c}{2})(\frac{3a-2a}{2})} \) [Note: properties of an equilateral triangle a=b=c]

=> \( \sqrt { \frac{3a}{2} (\frac{3a-2a}{2})(\frac{3a-2a}{2})(\frac{3a-2a}{2})} \)

=> \( \sqrt { \frac{3a}{2} (\frac{a}{2})(\frac{a}{2})(\frac{a}{2})} \) [Simplify]

=> \( \sqrt { \frac{3a^4}{16} } \)

=> \( \frac{\sqrt {3}}{4} a^2 \)

Proof Area of equilateral triangle = \( \frac{\sqrt {3}}{4} a^2 \)

**Example 1: ****What is the area of a triangle where every side is 4 cm long?**

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

=> \( \frac{\sqrt {3}}{4} 4^2 \)

=> \( 4 \sqrt {3} \) [Value of Root 3 =1.73]

=> \( 4 \times 1.73 = 6.92 cm^2 \)

**Right Triangle Formula **

In the case of the right triangle, one angle must be of 90 – degrees. As we discussed earlier, the sim of all three interior angles would be 180-degrees then the sum of the rest two angles should be 90-degree but it cannot be equal to 90-degree.

The right triangles are commonly used in Trigonometry and the Pythagorean theorem is also based on the same concept. The three sides of a Triangle in geometry are named as the base, hypotenuse, and the height.

Now when you know about the right-angled Triangle in detail, this is the time to calculate the area of the Right Triangle. The area is always measured in square units and defined as the space occupied by the two-dimensional shape within boundaries.

The area of a right triangle formula can be given as –

\[ Area \;of \;an \;Right\;Triangle = \frac{\sqrt{1}}{2}bh\]

\[ Perimeter \;of \;an \;Right \;Triangle = a+b+c\]

\[ semi\;Perimeter \;of \;an \;Right \;Triangle = \frac{a+b+c}{2}\]

where,

b is the Base of Right Triangle.

h is the Hypotenuse of Right Triangle.

a is the Hight of Right Triangle.

### Introduction of Triangle

In Geometry, we study a variety of shapes having unique properties that make them different from each other. One of the most common figures in the Triangle. This is a two-dimensional closed figure or a polygon having three sides, three vertices, and 3 angles. The sum of all three interior angles would be 180-degrees and the sum of all 3 exterior angles would be 360-degrees.

Further, a triangle is divided into multiple categories based on side length and angle. These are an equilateral, isosceles, scalene, and right-angled triangle. In this post, we will focus on the two most common types of triangle i.e. Equilateral Triangle and the Right Triangle.

If you would look into deep then based on a unique property of the Triangle, the sum of two sides will always be greater than the third side. In the figure below, we have given three popular types of triangles based on the angle. These are acute angle Triangle, Right Triangle, and the Obtuse Triangle.

**Type of Triangles based on Angles**

**Acute angle triangle**: If the angle between any two sides is less than 90-degrees then it is named as the acute angle triangle.**Right triangle**: When the angle between any two sides is equal to 90 it is called a right triangle. When the angle between any two sides is equal to 90-degree then this is named as the right-angled triangle or it is also known as the Right Triangle.**Obtuse angle triangle**: When the angle between any two sides is greater than 90 it is called an obtuse angle triangle. When the angle between any two sides is greater than 90-degree then it is named as the obtuse Triangle.