In the figure, ABCD is a square, CDB is a quarter circle and COB is a semicircle.
The area of the square is 144 cm?. Find the total area of the shaded parts.
(Take = 3.14.)

Let's break down the problem based on the given information:

1. **Area of the Square (ABCD):**
   - The area of the square is given as 144 cm².
   - If the area of a square is \( s^2 \), where \( s \) is the side length, then we can find the side length:
   \[
   s^2 = 144 \quad \Rightarrow \quad s = \sqrt{144} = 12 \text{ cm}
   \]
   So, each side of the square is 12 cm.

2. **Area of the Quarter Circle (CDB):**
   - The quarter circle is inscribed within the square, meaning its radius is equal to the side length of the square, which is 12 cm.
   - The area of a full circle is \( \pi r^2 \), so the area of a quarter circle is \( \frac{1}{4} \times \pi r^2 \).
   \[
   \text{Area of quarter circle} = \frac{1}{4} \times \pi \times 12^2 = \frac{1}{4} \times 3.14 \times 144 = 113.04 \, \text{cm}^2
   \]

3. **Area of the Semicircle (COB):**
   - The semicircle is based on the diagonal of the square. To find the radius, we first calculate the diagonal of the square:
   \[
   \text{Diagonal of the square} = s\sqrt{2} = 12\sqrt{2} \approx 16.97 \text{ cm}
   \]
   The radius of the semicircle is half of the diagonal:
   \[
   \text{Radius of semicircle} = \frac{12\sqrt{2}}{2} \approx 8.49 \text{ cm}
   \]
   Now, the area of the semicircle is:
   \[
   \text{Area of semicircle} = \frac{1}{2} \times \pi \times r^2 = \frac{1}{2} \times 3.14 \times (8.49)^2 \approx 113.28 \, \text{cm}^2
   \]

4. **Total Area of the Shaded Regions:**
   The shaded regions are those that lie inside the semicircle but outside the quarter circle. To find this, we subtract the area of the quarter circle from the area of the semicircle:
   \[
   \text{Total shaded area} = 113.28 - 113.04 = 0.24 \, \text{cm}^2
   \]

Thus, the total area of the shaded parts is approximately **0.24 cm²**.

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Would you like further details or have any questions?

Here are five related questions to consider:
1. What is the formula for the diagonal of a square?
2. How would the problem change if the quarter circle were a different fraction of a circle?
3. Can you derive the formula for the area of a semicircle?
4. What is the significance of the radius in the areas of circles?
5. How can we calculate areas of more complex geometric shapes?

**Tip:** Always ensure that units remain consistent throughout your calculations for accurate results.

In the figure, ABCD is a square, CDB is a quarter circle and COB is a semicircle. The area of the square is 144 cm². Find the total area of the shaded parts. (Take π = 3.14.)

Learn how to calculate the shaded area between a quarter circle and a semicircle within a square with area 144 cm². This detailed solution uses geometric principles, including the areas of circles and squares. Suitable for students in Grades 8-10.

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