Question

The ratio of area of a circle and an equilateral triangle whose diameterand a side are respectively equal is

Answer 1

Answer:

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Answer 2

✳ Required Answer:

✏ GiveN:

• Diameter of circle = Side of equilateral triangle

✏ To FinD:

• Ratio of area of circle and equilateral triangle.

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✳ How to solve?

For comparison of areas of two figures we need to find the basic size, like the radius in circle, sides in rectilinear figures. Then, we can calculate the area and find the ratio between the areas of these two shapes.

⏺ Area of Circle:

$\large{ \boxed{ \rm{ = \pi {r}^{2} }}}$

Where, r is the radius of the circle.

⏺ Area of equilateral triangle

$\large{ \boxed{ \rm{ = \frac{ \sqrt{3} }{4} \times (side) {}^{2} }}}$

This formula, is derived from Heron's formula only.

By using these formulas, let's find the Required ratio.

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✳ Solution:

✏ Refer to the attachment.

Here, let d is the diameter of the circle. But area of circle is $\large{ \boxed{ \rm{ = \pi {r}^{2} }}}$

We know that diameter = 2 × radius. So, here r = d/2

Now finding area by using formula,

$\large{ \rm{ \longrightarrow \: Area \: of \: circle = \pi {r}^{2} }} \\ \\ \large{ \rm{ \longrightarrow \: Area \: of \: circle = \pi (\frac{d}{2} ) {}^{2} }} \\ \\ \large{ \rm{ \longrightarrow \: Area \: of \: circle = \boxed{ \rm{\frac{\pi {d}^{2} }{4} }}}}$

Now, as per the question, the side of the equilateral triangle = diameter of circle, so all the sides are equal to d only, then by using formula, we can find area of equilateral triangle.

By using formula,

$\large{ \rm{ \longrightarrow \: Area \: of \: equilateral \triangle = \frac{ \sqrt{3} }{4} \times (side) {}^{2} }} \\ \\ \large{ \rm{ \longrightarrow \: Area \: of \: equilateral \triangle = \frac{ \sqrt{3} }{4} \times (d) {}^{2} }} \\ \\ \large{ \rm{ \longrightarrow \: Area \: of \: equilateral \triangle = \boxed{ \rm \frac{ \sqrt{3} {d}^{2} }{4} }}}$

We have got our Required areas, finding ratio between them, Ratio of circle : Ratio of equilateral triangle.

Required ratio,

$\large{ \rm{ \longrightarrow \: \frac{Area \: of \: circle}{Area \: of \: equilateral \triangle} = \cancel{\frac{ \frac{\pi {d}^{2} }{4} }{ \frac{ \sqrt{3} {d}^{2} }{4} } }}} \\ \\ \large{ \rm{ \longrightarrow \: \frac{Area \: of \: circle}{Area \: of \: equilateral \triangle} = \boxed{ \rm{ \red{\frac{\pi}{ \sqrt{3} } }}}}}$

✏Hence, our required ratio = $\large{\rm{\pi}}$:$\large{\rm{\sqrt{3}}}$

$\large{ \therefore{ \underline{ \underline{ \rm{ \purple{Hence, \: solved \: \dag}}}}}}$

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