Question

1.(a) the radii of two circles are 8cm and 6cm respectively. find the radius of the circle having its area equal to the sum of the areas of the two areas

b) find the area of circle inscribed in an square of side of 14cm.​

Answer 1

$\large\underline{\sf{Solution-}}$

Given that,

• Radius of first circle, $\rm \: r_1$= 8 cm

• Radius of second circle, $\rm \: r_2$= 6 cm

Let assume that

• Radius of required circle be r cm

According to statement,

$\rm \: Area_{(Circle)} = Area_{( {1}^{st} \: Circle)} + Area_{( {2}^{nd} \: Circle)}$

$\rm \: \pi {r}^{2} = \pi {r_1}^{2} + \pi {r_2}^{2}$

$\rm \: \pi {r}^{2} = \pi ({r_1}^{2} + {r_2}^{2})$

$\rm \: {r}^{2} = {r_1}^{2} + {r_2}^{2}$

On substituting the values of $\rm \: r_1,\:r_2$, we get

$\rm \: {r}^{2} = {8}^{2} + {6}^{2}$

$\rm \: {r}^{2} = 64 + 36$

$\rm \: {r}^{2} = 100$

$\rm \: {r}^{2} = {10}^{2}$

$\bf\implies \:r \: = \: 10 \: cm$

$\large\underline{\sf{Solution-b}}$

Given that, circle is inscribed in a square of side 14 cm.

So, it means diameter of circle is equals to side of square.

So, Diameter of circle = 28 cm

So, Radius of circle, r = 14 cm

Now,

$\rm \: Area_{(Circle)} \: = \: \pi \: {r}^{2}$

$\rm \: = \: \dfrac{22}{7} \times 14 \times 14$

$\rm \: = \: 22 \times 2 \times 14$

$\rm \: = \: 44 \times 14$

$\rm \: = \: 616 \: {cm}^{2}$

Hence,

$\bf\implies \:Area_{(Circle)} \: = \: 616 \: {cm}^{2}$

$\rule{190pt}{2pt}$

Additional Information :-

$\begin{gathered}\begin{gathered}\boxed{\begin {array}{cc}\\ \dag\quad \Large\underline{\bf Formulas\:of\:Areas:-}\\ \\ \star\sf Square=(side)^2\\ \\ \star\sf Rectangle=Length\times Breadth \\\\ \star\sf Triangle=\dfrac{1}{2}\times Base\times Height \\\\ \star \sf Scalene\triangle=\sqrt {s (s-a)(s-b)(s-c)}\\ \\ \star \sf Rhombus =\dfrac {1}{2}\times d_1\times d_2 \\\\ \star\sf Rhombus =\:\dfrac {1}{2}d\sqrt {4a^2-d^2}\\ \\ \star\sf Parallelogram =Base\times Height\\\\ \star\sf Trapezium =\dfrac {1}{2}(a+b)\times Height \\ \\ \star\sf Equilateral\:Triangle=\dfrac {\sqrt{3}}{4}(side)^2\end {array}}\end{gathered}\end{gathered}$

Answer 2

Answer:

Solution No 1 :-

Given :

• The radii of two circles are 8 cm and 6 cm respectively.
• The area of circle is equal to the sum of the areas of the two areas.

To Find :-

• What is the radius of the circle.

Formula Used :-

$\clubsuit$ Area Of Circle Formula :-

$\bigstar \: \: \sf\boxed{\bold{Area_{(Circle)} =\: {\pi}r^2}}\: \: \: \bigstar$

where,

• π = Pie or 22/7
• r = Radius

Solution :-

❒ In case of first radii :

$\mapsto$ The radii of first circle is 8 cm.

So, the area of the circle will be :

$\implies \bf Area_{(First\: Circle)} =\: {\pi}r^2$

$\implies \sf Area_{(First\: Circle)} =\: {\pi} \times (8)^2$

$\implies \sf Area_{(First\: Circle)} =\: {\pi} \times (8 \times 8)$

$\implies \sf Area_{(First\: Circle)} =\: {\pi} \times 64$

$\implies \sf\bold{Area_{(First\: Circle)} =\: 64{\pi}}$

❒ In case of second radii :

$\mapsto$ The radii of second circle is 6 cm.

So, the area of the circle will be :

$\implies \bf Area_{(Second\: Circle)} =\: {\pi}r^2$

$\implies \sf Area_{(Second\: Circle)} =\: {\pi} \times (6)^2$

$\implies \sf Area_{(Second\: Circle)} =\: {\pi} \times (6 \times 6)$

$\implies \sf Area_{(Second\: Circle)} =\: {\pi} \times 36$

$\implies \sf\bold{Area_{(Second\: Circle)} =\: 36{\pi}}$

Now, according to the question :

$\leadsto$ The area of circle is equal to the sum of the areas of the two areas.

So,

$\footnotesize \implies \sf\bold{\underline{Area_{(Circle)} =\: Area_{(First\: Circle)} + Area_{(Second\: Circle)}}}$

$\implies \sf {\pi}r^2 =\: 64{\pi} + 36{\pi}$

$\implies \sf {\cancel{{\pi}}} r^2 =\: 100 {\cancel{{\pi}}}$

$\implies \sf r^2 =\: 100$

$\implies \sf r =\: \sqrt{100}$

$\implies \sf\bold{r =\: 10\: cm}$

$\therefore$ The radius of the circle is 10 cm .

Solution No 2 :-

Given :

• A circle inscribed in an square of side of 14 cm.

To Find :-

• What is the area of circle.

Solution :-

Given :

• Side of Square = 14 cm

As we know that :

$\diamond\: \: \sf\boxed{\bold{Side_{(Square)} =\: Diameter_{(Circle)}}} \: \: \: \diamond$

So, we have to find the radius :

Given :

• Diameter of Circle = 14 cm

According to the question by using the formula we get,

$\implies \sf\boxed{\bold{Radius =\: \dfrac{Diameter}{2}}}$

$\implies \sf Radius =\: \dfrac{14}{2}$

$\implies \sf Radius =\: \dfrac{7}{1}$

$\implies \sf\bold{Radius =\: 7\: cm}$

Now, we have to find the area of circle :

Given :

• Radius = 7 cm

According to the question by using the formula we get,

$\implies \sf\boxed{\bold{Area_{(Circle)} =\: {\pi}r^2}}$

$\implies \sf Area_{(Circle)} =\: \dfrac{22}{7} \times (7)^2$

$\implies \sf Area_{(Circle)} =\: \dfrac{22}{7} \times (7 \times 7)$

$\implies \sf Area_{(Circle)} =\: \dfrac{22}{7} \times 49$

$\implies \sf Area_{(Circle)} =\: \dfrac{1078}{7}$

$\implies \sf\bold{\underline{Area_{(Circle)} =\: 154\: cm^2}}$

$\therefore$ The area of the circle is 154 cm² .