Area Of Equilateral Triangle

Equilateral Triangle Definition:

There are mainly three types of triangles. These are the Scalene triangle, Equilateral triangle, and Isosceles. Out of these three triangles we are going to study about a triangle which has equal sides and angles.

A triangle which has all the three sides equal and all angles equal to 60° is called an equilateral triangle. All the angles in an equilateral triangle are congruent.

Area of an equilateral triangle:

  • Consider an equilateral triangle having sides equal to a.
Equilateral Triangle

Equilateral Triangle

  • We drop a straight line from the top triangle vertex  to the midpoint of the base of the triangle, thus dividing the base into two equal halves.
Area Of Equilateral Triangle

Area Of Equilateral Triangle

  • Now we cut along the straight line and move the other half of triangle to form the rectangle.
How to find Area of Equilateral Triangle?

How to find Area of Equilateral Triangle?

We are considering the length of the equilateral triangle to be ‘a’ and let the height of it be ‘h’

So the area of an equilateral triangle = Area of a rectangle = \(\frac{1}{2}\times a \times h\) …………. (i)

The half of the rectangle is a right-angled triangle as it can be seen from the figure above.

Thereby applying the Pythagoras Theorem

\(\Rightarrow (a)^{2} = (h)^{2}+ \left ( \frac{a}{2} \right )^{2}\)

\(\Rightarrow (h)^{2} = \left ( \frac{3}{4} \right )a^{2}\)

\(\Rightarrow h = \frac{\sqrt{3}}{2} a \) …………..(ii)

Substituting the value of (ii) in (i), we have:

Area of an Equilateral Triangle \(= \frac{1}{2} \times a \times \frac{\sqrt{3}}{2} a\)

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

Properties of Equilateral Triangle

An equilateral triangle is the one in which all three sides are equal. It is a special case of the isosceles triangle where the third side is also equal. In the triangle ABC, AB = BC = CA.

Properties of an equilateral triangle are:

  • It has 3 equal sides.
  • It has 3 equal angles.
  • Since the sum of the interior angles is 180 degrees, every angle of an equilateral triangle is 60 degrees.
  • It is a regular polygon with 3 sides.
  • The altitude and median from a vertex is the same single line.
  • The ortho-center and centroid of the triangle is the same point.
  • In an equilateral triangle, median, angle bisector, and altitude for all sides are all the same and are the lines of symmetry of the equilateral triangle.
  • The area of an equilateral triangle is  √3 a2/ 4
  • The perimeter of an equilateral triangle is 3a.

Equilateral Triangle Examples

Question 1: Find the area of an equilateral triangle whose perimeter is 12 cm?

Solution

Given: Perimeter of an equilateral triangle = 12 cm

As per formula: Perimeter of the equilateral triangle = 3a, Where a is the side of the equilateral triangle.

Step 1: Find the side of an equilateral triangle using perimeter.

3a = 12

a = 4

Side is 4 cm.

Step 2: Find area of an equilateral triangle using formula.

Area, A = √3 a2/ 4 sq units

= √3 (4)2/ 4

= 4√3

Therefore, area of given equilateral triangle is  4√3 cm2 

Question 2: What is the area of an equilateral triangle whose side is 8cm?

Solution: The area of the equilateral triangle = √3 a2/ 4 cm2

                                                                                  = √3 × (82)/ 4 cm2

                                                                                   = √3 × 16 cm2

                                                                                    = 16 √3 cm2

This is all about the area of an equilateral triangle. To know more about the other characteristics of an equilateral triangle and other geometrical figures, please do visit BYJU’S or download BYJU’S-The Learning App.

 

Practise This Question

Using the graph find how many of the fruit/ fruits have more sales on Monday than Tuesday.

Leave a Comment

Your email address will not be published. Required fields are marked *