Question

If the area of a circle inscribed in an equilateral triangle is 4cm Square,then what is the area of the triangle

Answer 1

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✤ Required Answer:

✒️ GiveN:

• A circle is inscribed inside a equilateral triangle.
• Area of the circle is 4 cm²

✒️ To FinD:

• Area of the triangle....?

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✤ How to solve?

We should know how to find the area of a circle when the radius is given to us. The formula for this:

• Area of the circle = πr²

And, Finding the area of the equilateral triangle:

• Area of the equilateral triangle = √3/4(side)²

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✤ Solution:

We have,

• Area of the circle = 4 cm²

That means,

➝ πr² = 4 cm²

➝ 22/7 r² = 4 cm²

➝ r² = 4 × 7/22 cm²

➝ r² = 14/11 cm²

➝ r = √14/11 cm

๑ Refer to the attachment...

The radius of the inscribed circle is r that is OD = √14/11 cm and Here O is the incentre of the equilateral triangle. We know that, Incentre is also the centroid of an equilateral triangle, So AD is the median.

• The centroid divides the median in 2:1 ratio. So, if OD = r, then AD = 2r + r = 3r

Finding length of AD,

➝ AD = 3r

➝ AD = 3√14/11 cm

Each angle of an equilateral triangle is 60°. So, If we will concentrate only on one of the smaller triangle like ABD, then we can find the length of AB(side) by using trigonometry.

Finding length of AB(side),

➝ Sin 60° = p/h

➝ Sin 60° = AD/AB

➝ √3 / 2 = 3√14/11 / AB

➝ AB = 2√42 / 11 cm

Now, finding the area of triangle,

➝ Area of the equi. △ = √3/4 × (side)²

➝ Area of the equi. △ = √3/4 × (2√42 / 11)²

➝ Area of the equi. △ = 42√3/121 cm²

❒ So, Final answer is:

• $\Large{ \underline{ \boxed{ \red{ \rm{ \frac{ 42 \sqrt{3} }{121} \: cm {}^{2} }}}}}$

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Answer 2

Step-by-step explanation:

Area of the circle = πr²

Area of the circle = 4 cm²

πr² = 4 cm²

22/7 r² = 4 cm²

r² = 4 × 7/22 cm²

r² = 14/11 cm²

r = √14/11 cm

Finding length of AD,

AD = 3r

AD = 3√14/11 cm

Finding length of AB(side),

Sin 60° = p/h

Sin 60° = AD/AB

√3 / 2 = 3√14/11 / AB

AB = 2√42 / 11 cm

Area of the equi. △ = √3/4 × (side)²

Area of the equi. △ = √3/4 × (2√42 / 11)²

Area of the equi. △ = 42√3/121 cm²

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