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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