Let there be an equilateral ABC

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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

Area of equilateral triangle - Part 2

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

Area of equilateral triangle - Part 3

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

Area of equilateral triangle - Part 4

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

 

 

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.