Question

The radius of two circles are 9cm and 12cm.Find the radius of a circle whose area is equal to the sum of the area of these two circles​

Answer 1

★ Procedure :

Here we'll first calculate the area of both circles and then we'll add both to find the sum of these two circles.

Then, by using the formula of area ,we'll form a suitable equation in order to calculate the radius of a circle whose area is equal to the sum of the area of these two circles.

Given:

• Radius of the first circle = 9 cm

• Radius of the second circle = 12 cm

To calculate:

• The radius of a circle whose area is equal to the sum of the area of these two circles.

Calculation:

~ Calculating the area of the 1st circle :

$\red{\bigstar} \boxed{\tt {{Area}_{(Circle)} = \pi {r}^{2} }} \\ \\ \\ \sf { \longrightarrow {Area}_{(Circle)} = \pi \times {(9)}^{2}} \\ \\ \\ \sf { \longrightarrow {Area}_{(Circle)} = 81 \pi}$

~ Calculating area of the second circle :

$\red{\bigstar} \boxed{\tt {{Area}_{(Circle)} = \pi {r}^{2} }} \\ \\ \\ \sf { \longrightarrow {Area}_{(Circle)} = \pi \times {(12)}^{2}} \\ \\ \\ \sf { \longrightarrow {Area}_{(Circle)} = 144 \pi}$

~ Calculating area of both circles :

$\red{\bigstar} \boxed{\tt {{Area}_{(Both \: circles)} = {Area}_{(1st \: circle)} + {Area}_{(2nd \: circle)} }} \\ \\ \\ \sf { \longrightarrow {Area}_{(Both \: circles)} = {\pi{r}^{2}}_{(1st \: circle)} + {\pi {r}^{2}}_{(2nd \: circle)} } \\ \\ \\ \sf { \longrightarrow {Area}_{(Both \: circles)} = 81 \pi + 144 \pi} \\ \\ \\ \sf { \longrightarrow {Area}_{(Both \: circles)} = 225 \pi}$

~ Calculating the radius of a circle whose area is equal to the sum of the area of these two circles :

Let the radius be " r ".

$\red{\bigstar} \boxed{\tt {{Area}_{(Circle)} = \pi {r}^{2} }} \\ \\ \\ \sf { \longrightarrow 225 \pi = \pi \times {r}^{2}} \\ \\ \\ \sf { \longrightarrow \dfrac{225 \pi}{\pi }= {r}^{2} } \\ \\ \\ \sf { \longrightarrow 225= {r}^{2} } \\ \\ \\ \sf { \longrightarrow \sqrt{225}= r } \\ \\ \\ \longrightarrow \underline{\boxed{\sf{15 \: cm = r}}} \: \: \red{\bigstar}$

Henceforth, the radius of the circle 15 cm whose area is equal to the sum of the area of these two circles.

Answer 2

Given : Radius of two circles are 9cm and 12cm.

To Find : Radius of a circle whose area is equal to the sum of the area of these two circles.

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Concept :

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We have to find the areas of first and second circle. Then, we've to sum them. Then we'll have the area of big circle. After finding sum of those areas equate that sum with area of the circle formula to find it's radius.

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Let

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• Radius of first circle = R1
• Radius of second circle = R2
• Radius of largest circle = R

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★ Firstly, find the area of the first circle.

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$\sf:\implies Area_{(first \: circle)} = \bold \pi {R1 {}^{2} } \\ \\ \\ \sf:\implies Area_{(first \: circle)} = \pi \times 9 \times 9 \\ \\ \\ \sf:\implies \: \underline{\boxed{\mathfrak\purple{Area_{(first \: circle)} = 81\pi \: cm {}^{2} }}} \: \bigstar \:$

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★ Now, find the area of the second circle.

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$\sf:\implies Area_{(second\: circle)} = \bold \pi {R2 {}^{2} } \\ \\ \\ \sf:\implies Area_{(second\: circle)} = \pi \times 12 \times 12\\ \\ \\ \sf:\implies \: \underline{\boxed{\mathfrak\purple{Area_{(second \: circle)} = 144\pi \: cm {}^{2} }}} \: \bigstar \:$

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★ Now, let's find the area of largest circle.

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$\sf:\implies \: Area_{(largest\: circle)} = Area_{(first\: circle + second \: circle )} \\ \\ \\ \sf:\implies \: Area_{(largest\: circle)} = 81\pi + 144\pi \\ \\ \\ \sf:\implies \: \underline{\boxed{\mathfrak\purple{Area_{(largest\: circle)} = 225\pi \: {cm}^{2} }}} \: \bigstar \:$

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★ Now, put the value of area of the largest circle in the area of the circle formula.

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$\sf:\implies \: Area_{(largest\: circle)} \: = \bold \pi {R {}^{2} } \\ \\ \\ \sf:\implies 225\pi = \pi \:R {}^{2} \\ \\ \\ \sf:\implies \: R {}^{2} \: = 225\\ \\ \\ \sf:\implies \: R \: = \sqrt{225} \\ \\ \\ \sf:\implies \: \underline{\boxed{\mathfrak\purple{R =15cm }}} \: \bigstar \\ \\ \\ \\ \sf\therefore{\underline{Hence, \: the \: radius \: of \: the \: largest \:circle \: is \: \bold{15cm}. }}$