Question

find the area of a rectangle whose length is 10 cm and the perimeter is equal to the perimeter of square whose side is equal to the radius of a circle of circumference 154 cm​

Answer 1

Given:

• Length of rectangle = 10 cm.

• Perimeter = Perimeter of square.

• Side of square = Radius of the circle.

• Circumference of circle = 154 cm.

To calculate:

• Area of the rectangle.

Calculation:

From the given question, we can simply say that everything is linked. As the length of the rectangle is given, so we need to calculate the breadth in order to calculate area. To find breadth, we need its perimeter and to find perimeter, we need the perimeter of square. To find perimeter of square, we have to calculate its side. And, to find side of square , we need to calculate the radius of the circle whose circumference is 154 cm. So, let us solve it step by step !!

$\underline{\small{\boxed{\rm{\purple{ Calculating \: Radius \: of \: the\: circle :}}}}}$

Given : Circumference is 154 cm.

As we know that,

$\sf {\dashrightarrow C = 2\pi r}$

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$\sf {\dashrightarrow 154 \: cm = 2 \times \dfrac{22}{7} r \: cm}$

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$\sf {\dashrightarrow 154 \: cm = \dfrac{44}{7} r \: cm}$

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$\sf {\dashrightarrow 154 \times 7 \: cm = 44 r \: cm}$

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$\sf {\dashrightarrow 1078 \: cm = 44 r \: cm}$

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$\sf {\dashrightarrow \dfrac{1078}{44} \: cm = r \: cm}$

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$\boxed{ \sf {\dashrightarrow 24.5 \: cm = r \: cm}}$

Radius is 24.5 cm.

This means that,

• Side of the square is 24.5 cm.

Now, let us calculate the perimeter of the square to calculate the breadth of the rectangle.

$\underline{\small{\boxed{\rm{\purple{ Calculating \: perimeter \: of \: the\: square :}}}}}$

As we know that,

$\sf {\dashrightarrow {Perimeter}_{(Square)} = 4 \times side}$

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$\sf {\dashrightarrow {Perimeter}_{(Square)} = 4 \times 24.5 \: cm}$

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$\boxed{ \sf {\dashrightarrow {Perimeter}_{(Square)} = 98 \: cm} }$

Perimeter of the square is 98 cm.

Now , as the question states that perimeter of the rectangle is equal to the perimeter of square. So, now let us calculate the breadth of the perimeter by forming a suitable equation.

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$\underline{\small{\boxed{\rm{\purple{ Calculating \: breadth \: of \: the\: rectangle :}}}}}$

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$\sf {\dashrightarrow {Perimeter}_{(Rectangle)} = {Perimeter}_{(Square)}}$

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$\sf {\dashrightarrow 2(l+b) \: cm = 98 \: cm}$

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$\sf {\dashrightarrow 2l + 2b\: cm = 98 \: cm}$

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$\sf {\dashrightarrow 2(10) + 2b\: cm = 98 \: cm}$

As, length is 10 cm. [Given]

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$\sf {\dashrightarrow 20 + 2b \: cm = 98 \: cm}$

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$\sf {\dashrightarrow 2b \: cm = 98-20 \: cm}$

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$\sf {\dashrightarrow 2b \: cm = 78 \: cm}$

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$\sf {\dashrightarrow b \: cm = \dfrac{78}{2} \: cm}$

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$\boxed{\sf {\dashrightarrow b \: cm = 39 \: cm}}$

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• Hence, breadth is 39 cm.

Now, final step is we have to calculate its area.

$\underline{\small{\boxed{\rm{\red{ Calculating \: area \: of \: the\: rectangle :}}}}}$

As we know that,

$\sf {\dashrightarrow {Area}_{(Rectangle)} = length \times breadth}$

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$\sf {\dashrightarrow {Area}_{(Rectangle)} = 10 \times 39 \: {cm}^{2} }$

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$\boxed{ \sf \red{\dashrightarrow {Area}_{(Rectangle)} = 390 \: {cm}^{2} } }$

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Therefore, area of the rectangle is $\tt \orange{390 \: {cm}^{2}}$.

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Required Answer:

• Area of the rectangle ⇒ $\bf { 390 \: {cm}^{2} }$

Answer 2

Perimeter of rectangle = Perimeter of square

Circumference of circle = 154 cm

If radius = $r$, ∴ $C = 2 \pi r$

Since radius = Side of square,

⇒ $2 \pi s = 154 cm$

⇒ $s = \frac{ 154}{2 \pi } = 24.5 cm$

(taking $\pi = \frac{22}{7}$)

Perimeter = 4 × 24.5 cm = 98 cm

(perimeter of sq. = $4a$)

Let breadth be = $b$

So, $2(l + b) = 98$

⇒ $l + b = 49 cm$ ⇒ $b = 49 - 10 = 39 cm$

∵ Area of rectangle = $lb$

⇒ Area = $(10 \ cm)(39 \ cm) = 390 \ cm^2$