Question

The figure below is made from a square, an equilateral triangle, and three semicircles (purple, yellow and green). One semicircle (purple) has an area of 3.

What are the areas of the other two semicircles?​

Answer 1

Answer

Area of yellow semicircle = 4

Area of green semicircle = 8

Explanation

Given that the area of purple circle = 3 sq. units

Let, the radius of purple semicircle be r

We know that area of semicircle = πr²/2

Thus,

$\sf \pi\frac{r^2}{2} = 3$

This gives

$\sf r^2 = 3\times\frac{2}{\pi}$

$\sf r = \sqrt{\frac{6}{\pi}}$

Now, in the diagram it's given that the diameter of semicircle corresponds with the altitude of equilateral triangle.

That is, diameter of semi circle = altitude of equilateral triangle

We know that altitude of equilateral triangle is given as

$\sf \frac{\sqrt{3}}{2}a$ , where a is the side of triangle

This gives

$\sf \frac{\sqrt{3}}{2}a = 2\times r$

$\sf \frac{\sqrt{3}}{2}a = 2\times \sqrt{\frac{6}{\pi}}$

This gives

$\sf a = 4\sqrt{\frac{2}{\pi}}$

Now, we got the side of equilateral triangle which corresponds with side of square

→ side of equilateral triangle = side of square.

side of equilateral triangle = side of square. Side of square = diameter of semicircle yellow

→ diameter of yellow semicircle = $\sf 4\sqrt{\frac{2}{\pi}}$

This give radius as $\sf 2\sqrt{\frac{2}{\pi}}$

So, area of semicircle = πr²/2

→ area of semicircle yellow = $\sf \pi \times 4\times \frac{2}{\pi} \times \frac{1}{2}$

area of semicircle yellow = 4

Again, diameter of semicircle green is the diagonal of square.

Diagonal of square = a√2

→ diagonal of square = $\sf 4\sqrt{\frac{2}{\pi}} \times \sqrt{2}$

= $\sf 4\times \frac{2}{\sqrt{\pi}}$

So, this is the diameter of green semicircle

This means radius of green semicircle = $\sf \frac{4}{\sqrt{\pi}}$

So area of green semicircle = πr²/2

→ $\sf \pi \times \frac{16}{\pi} \times {2}$

area of green semicircle = 8

Answer 2

||✪✪ GIVEN ✪✪||

• Area of semicircle with purple = 3 unit².

|| ★★ FORMULA USED ★★ ||

• Height of Equaliteral ∆ = (√3/2) * side.
• Diagonal of Square = √2 * side.
• Area of semi-circle Depends on Its Diameter or radius Length .

|| ✰✰ ANSWER ✰✰ ||

❁❁ Refer To Image First .. ❁❁

Since , Basic Method is Little bit lengthy and Already solved , Let Try to Solve it with Ratio method Now.

Points To Remember First :-

→ From image we can see That, Let side of Square is a unit .

→ since , Equilateral ∆ is on one of the side of Square , so, each side of Equaliteral ∆ also a unit.

→ Diagonal of Square is √2 times of its side . That means we can say that, Diameter of Green circle is √2a unit.

→ Diameter of orange circle is one of the side of Square, That means, diameter of orange circle is a unit.

→ Now, Purple circle diameter is Equal to the Height of Equaliteral ∆ , That is Equal to (√3/2) Times of its side.

So, Diameter of Purple circle is (√3/2)a unit.

Now, we have Diameter of all 3 circles in respect to side of Square .

So,

→ Ratio of Diameter of = Purple : Orange : Green

☛ (√3/2 * a) : ( a ) : (√2a)

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Now, we know That, Ratio of Areas of circle , depends upon Diameter or Radius Length, or we can say That, If Three or more circles have Radius or diameter as a, b & c. Than Ratio of Their Area will be = a² : b² : c². [ Reason , π will be cancel, and if diameter than 2 from denominator will also be cancel ].

with This we can now solve Both Areas Easily without any calculation .

Since, Ratio of Diameter of Purple , orange and green is (√3/2 * a) : ( a ) : (√2a)

So, Ratio of Their Area will be :-

☛ (√3/2 * a)² : ( a )² : (√2a)²

☛ (3/4 a²) : (a²) : (2a²)

☛ 3a² : 4a² : 8a²

_____________________________

Now, we have Given That, Area of Purple circle is 3unit².

So,

☞ 3a² = 3 unit²

☞ 4a² = 4 unit² = orange circle Area.

☞ 8a² = 8 unit² = Green circle Area.

Hence, we can say That, Area of orange circle is 4 unit² and Area of Green circle is 8 unit².

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