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In the below figure, the square $ A B C D $ is divided into five equal parts, all having same area. The central part is circular and the lines $ A E, G C, B F $ and $ H D $ lie along the diagonals $ A C $ and $ B D $ of the square. If $ A B=22 \mathrm{~cm} $, find the circumference of the central part."


Given:

The square \( A B C D \) is divided into five equal parts, all having same area.

The central part is circular and the lines \( A E, G C, B F \) and \( H D \) lie along the diagonals \( A C \) and \( B D \) of the square.

\( A B=22 \mathrm{~cm} \).

To do: 

We have to find the circumference of the central part.

Solution:

Length of the side of the square $= 22\ cm$

This implies,

Area of the square $= (22)^2$

$= 484\ cm^2$

The square is divided into five parts equal in area.

This implies,

Area of each part $=\frac{484}{5}\ cm^2$

Area of the inside circle $=\frac{484}{5} \mathrm{~cm}^{2}$

Let the radius of the circle be $r$.

Therefore,

$\frac{484}{5}=\frac{22}{7}\times r^2$

$\Rightarrow r=\sqrt{\frac{484 \times 7}{5 \times 22}}$

$\Rightarrow r=\sqrt{\frac{22 \times 7}{5}}$

$\Rightarrow r=\sqrt{\frac{154}{5}}$

$\Rightarrow r=\sqrt{30.8}$

$\Rightarrow r=5.55 \mathrm{~cm}$

Circumference of the circle $=2 \pi r$

$=2 \times \frac{22}{7} \times 5.55 \mathrm{~cm}$

$=\frac{244.2}{7} \mathrm{~cm}$

$=34.88 \mathrm{~cm}$

The circumference of the central part is $34.88\ cm$.

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Updated on: 10-Oct-2022

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