As shown in the figure, there is a square of side 24 cm. A circle is inscribed inside the square. Inside the circle are four circles of equal radius which are inscribed. The total area of the shaded region in the figure given below is ________
Show Hint
Break the shaded area into square minus big circle plus small circles minus their overlaps.
Step 1: Note the circle sizes from the figure.
The square has side 24 cm, so the circle inscribed in it has radius 12 cm. The four equal circles inscribed inside that big circle, arranged like petals, each have radius 6 cm, based on the figure.
Step 2: Get the plain areas first, ignoring overlaps.
Square area $= 24^2 = 576$. Big circle area $= \pi(12)^2 = 144\pi$. The 4 small circles together give $4 \times \pi(6)^2 = 144\pi$.
Step 3: Use the circular segment formula for the overlap between two neighboring small circles.
Where two adjacent small circles overlap, the triangle formed by the two centers and a crossing point is a right triangle with both legs equal to the radius, 6 cm, so the angle at each center covering the overlap is $90^{\circ} = \frac{\pi}{2}$ radians. The area of one circular segment is $\frac{1}{2}r^2(\theta - \sin\theta) = \frac{1}{2}(36)\left(\frac{\pi}{2} - 1\right) = 9\pi - 18$. Each lens shaped overlap is made of two such segments, one from each circle, so one lens $= 2(9\pi-18) = 18\pi - 36$. With 4 such neighboring pairs around the flower pattern, the total overlap correction is $4(18\pi-36) = 72\pi - 144$.
Step 4: Combine and check against the options.
Shaded area $= 576 - 144\pi + 144\pi - (72\pi - 144) = 720 - 72\pi$, matching the other method exactly. Since this does not equal $576-196\pi$, $584-196\pi$ or $864-196\pi$, it points to none of the above.
Final Answer:
The shaded region works out to $720 - 72\pi$ sq cm, which is not listed, so option D is correct.
\[ \boxed{720 - 72\pi \Rightarrow \text{None of the above}} \]
