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Area of An Equilateral Triangle

Equilateral Triangle

  • An equilateral triangle is a triangle in which all three sides are equal.
  • Equilateral triangles are also equi-angular, which means, all three internal angles are also equal to each other and the only value possible is 60° each.
  • The area of an equilateral triangle is basically the amount of space occupied by an equilateral triangle.
  • Area of a triangle is measured in $unit^{2}$.
  • In an equilateral triangle, the median, angle bisector and perpendicular are all the same and can be simply termed as the perpendicular bisector due to congruence conditions.
  • A triangle is equilateral if and only if any three of the smaller triangles have either the same perimeter or the same inradius.
  • A triangle is equilateral if and only if the circumcenters of any three of the smaller triangles have the same distance from the centroid.

Area of Equilateral Triangle Formula:  A = $\frac{\sqrt{3}}{4}$$a^{2}$

where, “a” denoted the sides of an Equilateral Triangle

Proof:

Area of an Equilatereal triangle

In the figure above, the sides of an equilateral triangle are equal to “a” units.

We know that the area of Triangle is given by;

A = \frac{1}{2} \times base \times height

 

To find the height, consider Triangle ABC,
Applying Pythagoras Theorem we know,

AB^{2} = AD^{2}+BD^{2}

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

h^{2} = a^{2} - \frac{a^{2}}{4}

h^{2} = \frac{3a^{2}}{4}

h = \frac{\sqrt{3}a}{2}

Thus, we can calculate area by the basic equation,
A = \frac{1}{2} \times b \times h = \frac{1}{2} \times a \times \frac{\sqrt{3}a}{2}
Therefore, A = = \frac{\sqrt{3}a^{2}}{4} \;\; unit^{2}

  Lets work out a few examples:-

 

  Example 1: Find the area of an equilateral triangle whose side is 7 cm ?

  Solution:

  Given,

  Side of the equilateral triangle = a = 7 cm

  Area of an equilateral triangle =  $\frac{\sqrt{3}}{4}$ $a^{2}$

  = $\frac{\sqrt{3}}{4}$$\times$$7^{2}$ $cm^{2}$

  = $\frac{\sqrt{3}}{4}$$\times$49 $cm^{2}$

  = 21.21762 $cm^{2}$

 

  Example 2: Find the area of an equilateral triangle whose side is 28 cm ?

  Solution:

  Given,

  Side of the equilateral triangle (a) = 28 cm

  We know, Area of an equilateral triangle =  $\frac{\sqrt{3}}{4}$ $a^{2}$

  = $\frac{\sqrt{3}}{4}$$\times$$28^{2}$ $cm^{2}$

  = $\frac{\sqrt{3}}{4}$$\times$784 $cm^{2}$

  = 339.48196 $cm^{2}$

For more, Formulas of triangles.

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