Area of equilateral triangle

Let there be an equilateral ABC

We need to find its area

We know that,

  Area ∆ABC = 1/2 × Base × Height

Finding base & height of equilateral triangle ABC

Height is perpendicular from the vertex to the base.

Let us draw perpendicular from point A

So,

      Height = AD

      Base = BC = a

So, we need to find height AD

In equilateral triangle,

altitude is also the median

So, point D is also the mid-point of BC

Therefore,

  BD = DC = a/2

Now, in ∆ADC

By Pythagoras theorem

  AC ² = AD ² + DC ²

  a ² = AD ² + (a/2) ²

a ² = AD2 + a ^2/4

      AD^2 + a2/4 = a ²

     AD^2 = a2- a ^2/4

    AD^2 = (4a ²   - ^a2   )/4

   AD^2 = ^(3a2 )/4

  AD = √((3a ²   )/4)

  AD = a/2 √3

  AD = (√3  a)/2

Now,

  Height = AD

= √3/2 a

  Base = BC

= a

Area of ∆ABC = 1/2 × Base × Height

= 1/2 × a × √3/2 a

= √3/4 a^2

∴ Area of equilateral triangle = √3/4 a ²

Find area of the following equilateral triangle whose sides are 2 cm

Side = a = 2 cm

Area of equilateral ∆ABC = √3/4 a ²

= √3/4 (2) ²

= √3/4 × 4

= √3 cm ²

∴ Area of equilateral triangle ∆ABC is √3 cm ²