The diagram shows three circles, each of radius 4 cm - Edexcel - GCSE Maths - Question 3 - 2022 - Paper 1

The area of the isosceles triangle can be found using the formula:

$ext{Area} = \frac{1}{2} \times b \times h$

Where b is the base (which is the distance between points B and C, 4 cm), and h is the height from A to the base BC. Since the height forms another triangle where the height is equivalent to the distance from the center of the circle to the midpoint of AC.

Using the Pythagorean theorem, the height can be computed as:

$h = \sqrt{(4)^2 - (2)^2} = \sqrt{16 - 4} = \sqrt{12} = 2\sqrt{3}$

Thus, the area is:

$\text{Area} = \frac{1}{2} \times 4 \times 2\sqrt{3} = 4\sqrt{3} \text{ cm}^2$

To find the area of one shaded segment, use:

$\text{Area of Segment} = \text{Area of Sector} - \text{Area of Triangle}$

Hence,

$\text{Area of Segment} = \frac{16\pi}{3} - 4\sqrt{3}$

Since there are two identical shaded segments:

$\text{Total Area of Shaded Regions} = 2 \times \left(\frac{16\pi}{3} - 4\sqrt{3}\right) = \frac{32\pi}{3} - 8\sqrt{3} \text{ cm}^2$