In this image, we are tasked with calculating the area of a shaded region consisting of semicircles and a full circle, where the diameters of the semicircles are given in relation to one another. Let's break down the problem step by step:

Given Information:

• $SR = 4$ (The diameter of the smallest semicircle)
• $RQ = 2 \times SR = 8$ (The diameter of the middle semicircle)
• $QP = 2 \times RQ = 16$ (The diameter of the largest semicircle)

Step 1: Understanding the geometry

• The shaded region is made of three semicircles with diameters $SR = 4$, $RQ = 8$, and $QP = 16$.
• These semicircles are aligned in such a way that they form a figure with one large semicircle and two smaller ones inside.

Step 2: Calculating the area of the semicircles

The formula for the area of a semicircle is given by:

$A_{\text{semicircle}} = \frac{1}{2} \pi r^2$

where $r$ is the radius, which is half of the diameter.

• For the largest semicircle (diameter $16$): $r_1 = \frac{16}{2} = 8 \quad \Rightarrow \quad A_1 = \frac{1}{2} \pi (8)^2 = \frac{1}{2} \pi \times 64 = 32\pi$

• For the middle semicircle (diameter $8$): $r_2 = \frac{8}{2} = 4 \quad \Rightarrow \quad A_2 = \frac{1}{2} \pi (4)^2 = \frac{1}{2} \pi \times 16 = 8\pi$

• For the smallest semicircle (diameter $4$): $r_3 = \frac{4}{2} = 2 \quad \Rightarrow \quad A_3 = \frac{1}{2} \pi (2)^2 = \frac{1}{2} \pi \times 4 = 2\pi$

Step 3: Calculating the total shaded area

The shaded area consists of the largest semicircle minus the two smaller semicircles.

$\text{Shaded Area} = A_1 - (A_2 + A_3) = 32\pi - (8\pi + 2\pi) = 32\pi - 10\pi = 22\pi$

Thus, the area of the shaded region is:

$\boxed{22\pi}$

Do you need further clarification or additional details?

Here are 5 related questions:

1. What is the formula for the area of a full circle instead of a semicircle?
2. How would the solution change if the figures were full circles instead of semicircles?
3. Can you find the circumference of each semicircle using their radii?
4. How would you calculate the perimeter of the shaded region?
5. How would the area change if the diameter of the largest semicircle were doubled?

Tip:

When dealing with composite shapes, breaking them into simpler geometric figures can make complex area calculations much easier!