The diagram shows three circles, each of radius 4 cm - Edexcel - GCSE Maths - Question 1 - 2022 - Paper 2

To find the area of one shaded segment in circle A, we first need to determine the angle of the sector formed at center A. Since AB = BC = 4 cm and the radius of the circles is also 4 cm, triangle ABC is isosceles with sides AB and AC equal. The angle at A can be found using the cosine rule or recognizing symmetry.

The angle at A is 120 degrees (as angle B is formed by the two radii).

The area of the sector of circle A can be calculated as:

$\text{Area of sector} = \frac{120}{360} \times \pi r^2 = \frac{1}{3} \times \pi \times 4^2 = \frac{16}{3} \pi$

Next, we find the area of triangle ABC:

$\text{Area of triangle} = \frac{1}{2} \times base \times height$

For triangle ABC, the height can be calculated by using trigonometry, which gives the height as: $h = 4 \times \sin(60^\circ) = 4 \times \frac{\sqrt{3}}{2} = 2\sqrt{3}$

Thus, the area of triangle ABC is:

$\text{Area of triangle} = \frac{1}{2} \times 4 \times 2\sqrt{3} = 4\sqrt{3}$

Therefore, the area of the shaded segment in circle A is:

$\text{Area of segment} = \text{Area of sector} - \text{Area of triangle} = \frac{16}{3}\pi - 4\sqrt{3}$