Question

a wire in the form of square with side 22cm. it is reshaped and bent into the form of circle. calculate the area of this circle

Answer 1

Given : A a wire in the form of square with side 22cm. It is reshaped and bent into the form of circle .

To Find : Area of formed circle by wire .

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$\frak{\large{\underline{ Diagram }}}$

• Circle -:

$\setlength{\unitlength}{1cm}\begin{picture}(0,0)\thicklines\qbezier(2.3,0)(2.121,2.121)(0,2.3)\qbezier(-2.3,0)(-2.121,2.121)(0,2.3)\qbezier(-2.3,0)(-2.121,-2.121)(0,-2.3)\qbezier(2.3,0)(2.121,-2.121)(-0,-2.3)\put(0,0){\line(1,0){2.3}}\put(0.5,0.3){\bf\large 14 \ cm}\end{picture}$

• Square -:

$\setlength{\unitlength}{1cm}\begin{picture}(0,0)\thicklines\multiput(0,0)(4,0){2}{\line(0,1){4}}\multiput(0,0)(0,4){2}{\line(1,0){4}}\put(-0.5,-0.5){\bf D}\put(-0.5,4.2){\bf A}\put(4.2,-0.5){\bf C}\put(4.2,4.2){\bf B}\put(1.5,-0.6){\bf\large 22\ cm}\put(4.4,2){\bf\large 22\ cm}\end{picture}$

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❍ Finding Radius of formed circle:

$\underline{\frak{\dag\: As \:We\:know \:that \: ,}}$

⠀⠀⠀⠀⠀ $\underline {\boxed {\sf{ Perimeter _{(Square)} = 4 \times a \:units .}}}$

⠀⠀Here a is the length of side of Square in cm .

⠀⠀⠀⠀⠀ $\underline {\boxed {\sf{ Circumference _{(Circle)} = 2 \pi r\:units .}}}$

⠀⠀Here r is the Radius of Circle in cm and $\pi = \dfrac{22}{7}$ .

Given that ,

⠀⠀⠀⠀⠀A wire in the form of square and it is reshaped and bent into the form of circle.

Then ,

⠀⠀⠀⠀⠀ $\underline {\boxed {\sf{ Circumference _{(Circle)} = Perimeter _{(Square)} .}}}$

Or,

⠀⠀⠀⠀⠀ $\underline {\boxed {\sf{ 2 \times \pi \times Radius\:=4\times Side .}}}$

⠀⠀⠀⠀⠀⠀$\underline {\bf{\star\:Now \: By \: Substituting \: the \: Given \: Values \::}}$

⠀⠀⠀⠀⠀ $\implies \tt{ 2 \times \pi \times Radius\:=4\times 22 }$

⠀⠀⠀⠀⠀ $\implies \tt{ 2 \times \pi \times Radius\:=88 \:cm }$

⠀⠀⠀⠀⠀ $\implies \sf{ 2 \times \dfrac{22}{7} \times Radius\:=88\:cm }$

⠀⠀⠀⠀⠀ $\implies \sf{ Radius\:=\dfrac{88\times 7 }{2 \times 22} }$

⠀⠀⠀⠀⠀ $\implies \sf{ Radius\:=\dfrac{\cancel {616}}{\cancel {44}} }$

⠀⠀⠀⠀⠀$\underline {\boxed{\pink{ \mathrm {  Radius = 14\: cm}}}}\:\bf{\bigstar}$

Therefore,

⠀⠀⠀⠀⠀$\therefore {\underline { \mathrm {  Radius \:of\:Circle \:is\:14\: cm}}}$

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❍ Finding Area of Formed Circle:

$\underline{\frak{\dag\: As \:We\:know \:that \: ,}}$

⠀⠀⠀⠀⠀ $\underline {\boxed {\sf{ Area _{(Circle)} = \pi r^{2}\:sq.units .}}}$

⠀⠀Here r is the Radius of Circle in cm and $\pi = \dfrac{22}{7}$ .

⠀⠀⠀⠀⠀⠀$\underline {\bf{\star\:Now \: By \: Substituting \: the \: found \: Values \::}}$

⠀⠀⠀⠀⠀ $\implies \tt{Area_{(Circle)} = \pi \times 14^{2} \: }$

⠀⠀⠀⠀⠀ $\implies \tt{Area_{(Circle)} = \pi \times 196 \: }$

⠀⠀⠀⠀⠀ $\implies \tt{Area_{(Circle)} = \dfrac{22}{\cancel {7}} \times \cancel {196} \: }$

⠀⠀⠀⠀⠀ $\implies \tt{Area_{(Circle)} = 22 \times 28 \: }$

⠀⠀⠀⠀⠀$\underline {\boxed{\pink{ \mathrm {  Area_{(Circle)} = 616\: cm^{2}}}}}\:\bf{\bigstar}$

Therefore,

⠀⠀⠀⠀⠀$\therefore {\underline { \mathrm {  Area \:of\:Formed\:Circle \:is\:616\: cm^{2}}}}$

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$\qquad \qquad \underline {\large{\bf{\bigstar\:More\:To\:Know\::}}}$

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

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Answer 2

${ \large{ \underline{ \pmb{ \frak{Given \: that}}}}}$

★ a wire in the form of square with side 22cm

★ it is re shaped and bent into the form of circle.

${ \large{ \underline{ \pmb{ \frak{To \: find}}}}}$

★The area of the circle

${ \large{ \underline{ \pmb{ \frak{Understanding \: the \: question}}}}}$

☀️Concept : here, it is said that a wire was bent to form a square of side 22cm and then rebent to form a circle which means length of wire used for making a square = length of wire used for making the circle. which is nothing but perimeter of the square equals perimeter of the circle

❍So,now we have to find the perimeter of the square as we know the perimeter of the square and the circle are the same. then we should find the radius of the circle to find its area with the help of the perimeter

${ \large{ \underline{ \pmb{ \frak{Let's \: use \: the \: concept}}}}}$

★Formula to find the perimeter of a square is that perimeter of a square is 4 times its side (According to the given measurements)

★ Formula to find the circumference of a circle is that perimeter of a circle is 2πr

★ Formula to find the area of a circle is that area of a circle is  πr²

${ \large{ \underline{ \pmb{ \frak{Solution}}}}}$

★ Area of the circle is 616cm²

${ \large{ \underline{ \pmb{ \frak{Full \: solution}}}}}$

~ Now firstly let's find the perimeter of the square and as we know that side of the square is 22cm let's substitute it in the formula

✪ Perimeter of a square is 4 times its side

➠ Perimeter of the square = 4 × side

➠ Perimeter of the square = 4 × 22cm

➠ Perimeter of the square = 88cm

• Therefore perimeter of the square is 88cm

~As we know that perimeter of the square = circumference of the circle so, circumference of the circle is 88cm Now, let's find its radius.

✪ Circumference of a circle = 2πr

➠ Circumference of the circle = 2 × 22/7 × radius

➠ 88cm = 44/7 × radius

➠ radius = 88×7/44

➠ radius = 616/44

➠ radius = 14cm

• Therefore radius of the circle is 14cm

~Now let's find the area of the circle as we have found its radius. putting the value of radius in the formula

✪ Area of a circle is πr²

➠ Area of the circle = 22/7 × 14²

➠ Area of the circle = 22/7 × 14 × 14

➠ Area of the circle = 616cm²

• Therefore the area of the circle is 616cm²

${ \large{ \underline{ \pmb{ \frak{More \: to \: know}}}}}$

★Area of a rectangle = length × breadth

★ Area of a square = side × side

★ Area of a triangle = ½ × base × hieght

★ Area of a rhombus = d₁ × d₂ / 2

★ Area of a parallelogram = base × hieght

★ Perimeter of a rectangle= 2(length + breadth)

${ \large{ \underline{ \pmb{ \frak{Diagrams}}}}}$

❍Square :

$\setlength{\unitlength}{1cm}\begin{picture}(0,0)\thicklines\multiput(0,0)(4,0){2}{\line(0,1){4}}\multiput(0,0)(0,4){2}{\line(1,0){4}}\put(-0.5,-0.5){\bf D}\put(-0.5,4.2){\bf A}\put(4.2,-0.5){\bf C}\put(4.2,4.2){\bf B}\put(1.5,-0.6){\bf\large 22\ cm}\put(4.4,2){\bf\large 22\ cm}\end{picture}$

❍Circle :

$\setlength{\unitlength}{1cm}\begin{picture}(0,0)\thicklines\qbezier(2.3,0)(2.121,2.121)(0,2.3)\qbezier(-2.3,0)(-2.121,2.121)(0,2.3)\qbezier(-2.3,0)(-2.121,-2.121)(0,-2.3)\qbezier(2.3,0)(2.121,-2.121)(-0,-2.3)\put(0,0){\line(1,0){2.3}}\put(0.5,0.3){\bf\large 14 cm}\end{picture}$

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