Mathematics

Equilateral Triangle

Triangles are closed figures joining with the straight line segments .there are different types of triangles from which the Equilateral Triangle is a regular polygon figure that has three equal-lengths congruent sides (same measure). hence their all three angles should also be equal (equiangular). as the internal angles are acute, the equilateral triangle is itself acute.

since, we knew that all the sides and angles are equal in length, so their sum will also be equal to 180º and each interior angle will be 60º.

Equilateral Triangle

What are the Elements of the Equilateral Triangle?

from the above figure, the following are elements of such type triangles:
Sides: AB, BC, and CA are three equal sides and measures a,b and c respectively.
Vertices: A, B, C are called Vertices.
Interior Angles: let ∝, β, and γ are three interior angles and are equal to 60º each.
Exterior Angles: let x,y,z are three exterior angles that are supleentary to their adjacent interior angles i.e., ∝ + x = β + y = γ + z = 180º.
points to remember:
  • for equilateral triangle, sides a = b = c.
  • ∝ = β = γ = 60º
  • x=y=z= 120º or obtuse (greater than 90º)

Equilateral Triangle Area

The area of ​​a flat figure like the Equilateral Triangle represents the size of its surface. the multiplication of base (b) with height (h) and dividing by 2 will give us the area of an equilateral triangle. in other words, the area will be one-half of the base (a) times its height.

Equilateral Triangle Area

2

let suppose we have a height √3a/2. by putting the value of height from the above figure we have the area an equilateral triangle:

final area of the equilateral triangle

The perimeter of an equilateral triangle

as the following triangle have three sides all equal to each other, the perimeter corresponds to the sum of all sides or three times the length of one of its sides ( a ).

Mathematically,

P = a + a + a = 3a

 

equilateral triangle height

to find the height of an equilateral triangle we use the Pythagorean theorem. for this purpose, let us divide the triangle into two parts to make a right triangle. now take the sides a, a / 2, and height h. where side a / 2 and h are known as legs while a is called the hypotenuse.

Equilateral triangle height

Applying the Pythagorean theorem
hence, we find the height (h) of the triangle = √3a/2.

What is the Pythagorean theorem?

this theorem states that two legs of the right triangle lie on the hypotenuse. the two sides that characteristic a right triangle are known as legs while the longest opposite side to legs is the hypotenuse.

according to the theorem:

the square of the hypotenuse is equal to the sum of the square of its legs.

c² = a² + b²

where a and b are two legs and c point to the hypotenuse.

Properties

The following properties fall under the Mentioned triangle:

  • Angle Bisector: 
  • Segment Bisector: 
  • median is that line that starts from mid of its any side and meets at the vertex of another opposite side. this means, we can draw three medians at a time and they will meet at a single point called the center of gravity.
  • axes of symmetry: it has three axes of symmetry that pass through the midpoint of each opposite side and its vertex
  • consists of three equal sides
  • known as a regular polygon having three sides.
  • The area is equal to √3a2/ 4
  • The perimeter is equal to 3a.
  • all the angles are 60º and are congruent to each other.
  • centroid and ortho-center meet at the same point.

Questions and Solved Excercise

Q. if the side of an equilateral triangle is 12cm side, find its area?
Ans: since,
   A=L² √3/2
   A=12² √3/2
   A=144 √3/2
   A= 72 √3 cm²
Q. let the sides of a triangle are equal to 5cm each. what will e it’s area?
we know that Area= √3/4.a²
Area= √3/4.5² = 10.8253175473 cm².
Q.  if the sides are sides 24cm long, calculate the area of a triangle.
Ans: since, area formula is given by:
  P = 3. L
  P = 3.24
  P = 72

FAQ’s (Frequently Asked Questions)

What is the equilateral triangle?

Ans: it is a regular polygon triangle that has three sides of equal lengths as well as equal angles. the internal angles are called congruent and are of 60º each.

What is equilateral triangle example?

Ans: rack in billiards, a slice of a pizza is an example of such type of triangle.

What is the area and perimeter of equilateral triangle?

Ans: the area is said to e √3 a2/ 4 while the perimeter is equal to 3a. where a is the side of an equilateral triangle.

What are the rules for an equilateral triangle?

Ans: basic properties and rules are given below:

  • it is a regular polygon in geometry.
  • All three sides are equal in length.
  • All three angles are congruent and equal to 60º.
  • The ortho-center and centroid meet at the same point.

What are the 3 main types of triangles?

Ans: equilateral, isosceles, and scalene are three major types of triangles based on their sides.

How many sides does an equilateral triangle have?

Ans: it has 3 sides all are equal in length.

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