Question

Which has greater area, a square of perimeter 44 cm or a circle having circumference 44 cm by the how much?​

Answer 1

Aɴsᴡᴇʀ

we have,

• perimeter of square = 44 cm
• circumference of circle = 44cm

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we know that,

$\boxed{ \bold{perimeter \: of \: square = 4 \times side}}$

$\bold{: \implies 44 = 4 \times side} \\ \\ \bold{ : \implies \: side = \cancel{ \frac{44}{4} }} \: \: \: \: \\ \\ \bold{ : \implies \: side = 11 \: cm}$

so, now we get side of square is 11cm

then,

$\boxed{ \bold{area \: of \: circle ={side}^{2} }}$

$\bold{ : \implies \: A1 = {(11\: cm)}^{2} } \\ \\ : \implies{ \underline{\boxed{ \bold{ A1 = 121 \: {cm}^{2} }}}}$

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we know that,

$\boxed{ \bold{circumference \: of \: circle = 2\pi r}}$

$\bold{ : \implies 44 = 2\pi r} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \bold{ : \implies \: 44 = 2 \times \frac{22}{7} \times r } \\ \\ \bold{ : \implies r = \frac{44 \times 7}{2 \times 22} } \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \bold{ : \implies r = \frac{ \cancel{44} \times 7}{ \cancel{44}} } \: \: \: \: \: \: \: \: \: \: \: \\ \\ \bold{ : \implies \: r = 7 \: cm} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

so, we get radius of the circle is 7 cm

then,

$\boxed{ \bold{ area \: of \: circle = \pi {r}^{2} }}$

$\bold{ : \implies A2 = \frac{22}{7} \times {7}^{2} } \: \: \: \: \: \: \: \\ \\ \bold{: \implies \: A2 = \frac{22}{ \cancel{7}} \times \cancel{ 7} \times 7} \\ \\ \bold{ : \implies A2 = 22 \times 7} \: \: \: \: \: \: \: \: \: \: \\ \\ : \implies{ \underline{\boxed{ \bold{ \:A2 =154 \: {cm}^{2} } }}} \: \:$

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we get,

• Area of square = 121 cm²
• Area of circle = 154 cm²

so, now compare the area of square to the area of circle

• 154 cm² - 121 cm² = 33 cm²

➪ A1 < A2

therefore, the area of circle is greater than the area of square by 33 cm²