Question

The height of an equilateral triangle is 6 cm. Find its area. [Take \[\sqrt 3 = 1.73\] ]

Answer 1

Hint: Here, height of the equilateral triangle is given, so using Pythagoras theorem we can find the base of the triangle. Now, we have base and height, using formula we can find the area of the triangle.

Complete step-by-step answer:
Drawn below diagram of an equilateral triangle ABC of height 6 cm.

As ABC is an equilateral triangle, AB = BC = AC.
D is the midpoint of BC, so, BD = DC = $ \dfrac{1}{2} $ BC
Also, AD is perpendicular to BC, therefore, ∠ADC = 90°.
In triangle ABC,
 $ A{C^2} = A{D^2} + D{C^2} $ [Pythagoras Theorem]
Putting AD = 6 cm, DC = BC/2 and AC = BC
 $ \Rightarrow B{C^2} = {6^2} + {\left( {\dfrac{{BC}}{2}} \right)^2} $
On simplifying
 $ \Rightarrow B{C^2} = 36 + \dfrac{{B{C^2}}}{4} $
 $ \Rightarrow B{C^2} - \dfrac{{B{C^2}}}{4} = 36 $
 $ \Rightarrow \dfrac{{3B{C^2}}}{4} = 36 $
 \[ \Rightarrow B{C^2} = \dfrac{{36 \times 4}}{3}\]
 \[ \Rightarrow BC = \dfrac{{6 \times 2}}{{\sqrt 3 }} = \dfrac{{12}}{{\sqrt 3 }}\]
 \[ \Rightarrow BC = \dfrac{{12}}{{\sqrt 3 }} \times \dfrac{{\sqrt 3 }}{{\sqrt 3 }} = \dfrac{{12}}{3} \times \sqrt 3 = 4\sqrt 3 \]
⇒ BC = 4 × 1.73 = 6.92 cm
Now, area of triangle = $ \dfrac{1}{2} $ × Base × Height
Here, Base = BC = 6.92 cm and Height = AD = 6 cm
Area of triangle ABC = $ \dfrac{1}{2} $ × 6.92 × 6 = 3 × 6.92 sq. cm
Area = 20.76 sq. cm
So, the correct answer is “20.76 sq. cm”.

Note: In these types of questions, we should have knowledge of some properties of equilateral triangles.
Here, height of the triangle is given so we can find the side of the triangle using Pythagoras Theorem. As the triangle is equilateral all sides are equal as well as all three heights with respect to sides are also equal.
Alternatively, we can find the side of a given equilateral triangle, if its height is given.
Height of equilateral triangle = $ \dfrac{{\sqrt 3 }}{2} $ × Side
⇒ 6 cm = $ \dfrac{{\sqrt 3 }}{2} $ × Side ⇒ Side = $ \dfrac{{12}}{{\sqrt 3 }} $
Now, area of equilateral triangle = $ \dfrac{{\sqrt 3 }}{4} \times {({\text{Side}})^2} $
Area = $ \dfrac{{\sqrt 3 }}{4} \times {\left( {\dfrac{{12}}{{\sqrt 3 }}} \right)^2} = 12\sqrt 3 = 12 \times 1.73 = 20.76 $ sq. cm.