Question: User: 25. The figure below is made up of 3 semicircles and a circle. OP = OQ and PQ = 42 cm. Find the total area of the shaded parts.
(Take \pi = 22/7.)
42 cm
Ans: ____________ cm²
top-left: Three semicircles and one circle arranged inside a large semicircle over diameter PQ; the left and right small semicircles are on PO and OQ, and the central circle overlaps them, with the shaded regions being the outer side areas and the two lens-shaped overlaps.
1. **Problem Statement:**
We have a figure composed of three semicircles and one circle arranged along the segment $PQ$ where $PQ = 42$ cm and $OP = OQ$. We need to find the total area of the shaded parts.
2. **Understanding the figure and variables:**
- Since $OP = OQ$, point $O$ is the midpoint of $PQ$, so $OP = OQ = \frac{PQ}{2} = \frac{42}{2} = 21$ cm.
- There is a large semicircle with diameter $PQ = 42$ cm.
- Two smaller semicircles are on $PO$ and $OQ$ each with diameter $21$ cm.
- A circle is inscribed overlapping the two smaller semicircles.
3. **Formulas and constants:**
- Area of a circle: $A = \pi r^2$
- Area of a semicircle: $A = \frac{1}{2} \pi r^2$
- Given $\pi = \frac{22}{7}$.
4. **Calculate areas of semicircles:**
- Large semicircle radius: $r_{large} = \frac{42}{2} = 21$ cm.
- Area of large semicircle:
$$
A_{large} = \frac{1}{2} \times \pi \times 21^2 = \frac{1}{2} \times \frac{22}{7} \times 441 = \frac{1}{2} \times \frac{22}{7} \times 441
$$
Calculate:
$$
\frac{22}{7} \times 441 = 22 \times 63 = 1386
$$
So,
$$
A_{large} = \frac{1}{2} \times 1386 = 693 \text{ cm}^2
$$
- Smaller semicircles radius: $r_{small} = \frac{21}{2} = 10.5$ cm.
- Area of one small semicircle:
$$
A_{small} = \frac{1}{2} \times \pi \times (10.5)^2 = \frac{1}{2} \times \frac{22}{7} \times 110.25
$$
Calculate:
$$
\frac{22}{7} \times 110.25 = 22 \times 15.75 = 346.5
$$
So,
$$
A_{small} = \frac{1}{2} \times 346.5 = 173.25 \text{ cm}^2
$$
- Total area of two small semicircles:
$$
2 \times 173.25 = 346.5 \text{ cm}^2
$$
5. **Calculate area of the circle in the middle:**
- The circle fits exactly between the two semicircles on $PO$ and $OQ$.
- Its diameter equals the radius of the large semicircle, which is $21$ cm.
- So radius of the circle:
$$
r_{circle} = \frac{21}{2} = 10.5 \text{ cm}
$$
- Area of the circle:
$$
A_{circle} = \pi r^2 = \frac{22}{7} \times (10.5)^2 = \frac{22}{7} \times 110.25 = 346.5 \text{ cm}^2
$$
6. **Calculate total shaded area:**
- The shaded area consists of the large semicircle minus the two small semicircles plus the circle in the middle (since the circle overlaps the small semicircles, the shaded area includes the circle).
- Total shaded area:
$$
A_{shaded} = A_{large} - 2 \times A_{small} + A_{circle} = 693 - 346.5 + 346.5 = 693 \text{ cm}^2
$$
7. **Final answer:**
$$
\boxed{693 \text{ cm}^2}
$$