Equilateral Triangle

Introduction

Equilateral triangle is a triangle in which all three sides have the same length . Triangle traces can be found all over the world, especially in architecture. They were well-liked by the ancient Egyptians. Triangles are divided into three types based on the length of their sides.

The triangles are isosceles, scalene, and equilateral. An Equilateral triangle is a triangle with three equal sides and angles. As each angle of an equilateral triangle is 60 degrees, it is also referred to as an equiangular triangle. The opposite side of an equilateral triangle is divided into equal lengths by a perpendicular drawn from any vertex. Additionally, the vertex angle is split into two equal parts, each 30 degrees away from the point where the perpendicular is drawn.

Equilateral Triangle

Triangle traces can be found all over the world, particularly in construction. They were popular among the ancient Egyptians. The faces of Egypt's famous Pyramids are triangular. The Giza Pyramids are the most well-known pyramids built by the Egyptians in the second century B.C. The Egyptians believed that the pyramid's triangular shape represented the earthly residence of their Sun God, Ra.

Triangles are classified into three types based on their sides. They are the isosceles, scalene, and equilateral triangles. The isosceles and scalene triangles are not the same as the equilateral triangle.

An Equilateral triangle is a triangle with three equal sides and angles. As each angle of an equilateral triangle is 60 degrees, it is also referred to as an equiangular triangle. Because the angles and sides of an equilateral triangle are equal, it is considered a regular polygon or a regular triangle.

Equilateral triangles are triangles with three equal sides, also known as regular triangles. Isosceles triangles are subsets of equilateral triangles, which have all three sides equal. An equilateral triangle has 3 equal sides.

All of the sides of the scalene triangle are not equal, and neither are the angles. Two sides of isosceles triangle are same.

Properties of Equilateral Triangle

An equilateral triangle has some properties that distinguish it from other triangles. To identify an equilateral triangle, use the properties listed below.

• An Equilateral triangle is a triangle with three equal sides

• As each angle of an equilateral triangle is 60 degrees, it is also referred to as an equiangular triangle.

• Because it has three sides, it is a regular polygon.

• A perpendicular drawn from any vertex to an equilateral triangle's opposite side bisects the side in equal lengths. It also divides the vertex angle into equal halves, each 30 degrees from where the perpendicular is drawn.

• The ortho-centre and centroid are both at the same location.

• The median, angle bisector, and altitude of an equilateral triangle are the same for all sides.

• Area of an equilateral triangle $\mathrm{=\:\frac{\sqrt{3}a^{2}}{4}}$, where a = equilateral triangle side

• The perimeter of an equilateral triangle is equal to 3a, where a is the side of an equilateral triangle.

• The sum of an equilateral triangle's angles equals 180 degrees.

Centroid of the equilateral triangle

The centroid of the equilateral triangle is located in the triangle's centre. Because all angles and sides are equal in length, finding the centroid is simple.

We must draw perpendiculars to each vertex of the triangle from the opposite sides. These must intersect at a common point, and the perpendiculars must all be equal, which is known as the centroid.

Circumcenter of an equilateral triangle

The circumcenter of an equilateral triangle is the point of intersection of perpendicular bisectors of the side. The circumcircle passes through all three vertices of the triangle.

An equilateral triangle is one in which the circumcenter of a triangle coincides with the orthocenter, incenter, or centroid.

Area of an Equilateral triangle

The area of an equilateral triangle is the space it occupies in a two-dimensional plane. Here is the formula for calculating the area of an equilateral triangle:

Area of an equilateral triangle $\mathrm{=\:\frac{\sqrt{3}a^{2}}{4}}$, where a = equilateral triangle side

Solved problems

1) If PQ = QR = RP = 2 cm, what is the area of the equilateral triangle PQR?

Answer − We will use the formula − $\mathrm{area\:=\:\frac{\sqrt{3}}{4}\:\times\:side^{2}}$

According to the question, side length = 2 cm.

As a result, $\mathrm{area\:=\:\frac{\sqrt{3}}{4}\:\times\:2^{2}}$

$\mathrm{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\sqrt{3}cm}$

2)What is the perimeter and semi-perimeter of an equilateral triangle with 10 units of side length?

Answer − We are aware that the perimeter of an equilateral triangle is three times the side length and that the semi-perimeter is half the perimeter.

side length = 10 units.

As a result, the equilateral triangle's perimeter will be $\mathrm{3\times\:10\:=\:30\:units}$

An equilateral triangle's semi-perimeter will be $\mathrm{\frac{30}{2}\:=\:15\:units.}$

3)What is the perimeter of an equilateral triangle if all of its sides are 30 inches long?

Answer − We are aware that the perimeter of an equilateral triangle is three times the length of its sides. The question specifies that side length $\mathrm{=\:30\:inches}$

As a result, the perimeter of an equilateral triangle is $\mathrm{3\times\:30\:=\:90\:inches}$

Conclusion

Triangle patterns are prevalent throughout the world, particularly in architecture. They enjoyed good favour with the Egyptians of antiquity. According to how long their sides are, triangles can be classified into three categories.

The triangles are equilateral, isosceles, and scalene. An Equilateral triangle is a triangle with three equal sides and angles. As each angle of an equilateral triangle is 60 degrees, it is also referred to as an equiangular triangle. The opposite side of an equilateral triangle is divided into equal lengths by a perpendicular drawn from any vertex. Additionally, the vertex angle is split into two equal parts, each 30 degrees away from the point where the perpendicular is drawn.

FAQs

1. How do you determine whether a triangle is equilateral?

If all of the sides of a triangle are the same length, it is said to be an equilateral triangle. When X, Y, and Z represent the three sides of a triangle, the triangle can only be equilateral if X equals Y equals Z. If one side of the triangle is equal to the other, then the triangle is said to be isosceles.

2. What kinds of triangles cannot exist?

A triangle cannot be both obtuse and equilateral. An obtuse angle cannot exist in an equilateral triangle because all three angles measure 60 degrees.

3. Is it possible to form a right equilateral triangle?

An equilateral triangle can never be a right triangle because one angle in a right triangle is 90 degrees, and if we make all angles equal to 90 degree by definition, their sum is 270 degrees, which is not possible because the sum of all triangle angles is 180 degree.

4. What are the applications of triangles in your life?

Trusses can be built using triangles. Trusses are used in a variety of structures, including roofs, bridges, and buildings. Trusses are triangles formed by combining horizontal and diagonal beams. Truss bridges are bridges that use trusses.