The diagram shows a square ABCD with sides of length 20cm - Edexcel - GCSE Maths - Question 10 - 2018 - Paper 1

The radius of the semicircle is half of AB, which is 10 cm.

The area of the semicircle is given by:

$\text{Area of semicircle} = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi (10)^2 = 50\pi \text{ cm}^2$

The radius of the circle with center B is equal to AC which is also 10 cm.

The area of this circle is:

$\text{Area of circle} = \pi r^2 = \pi (10)^2 = 100\pi \text{ cm}^2$

The shaded region is the area of the semicircle minus the area of the triangle formed by points A, C, and B.

To find the area of triangle ABC:

• Base AB = 20 cm
• Height from C to AB = 10 cm (length AC)

Thus,

$\text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 20 \times 10 = 100 \text{ cm}^2$

Now, the area of the shaded region:

$\text{Area of shaded region} = \text{Area of semicircle} - \text{Area of triangle} = 50\pi - 100$

Finally, we compare the areas:

• The area of the shaded region divided by the area of the square:

$\frac{50\pi - 100}{400} = \frac{\pi - 2}{8}$

To show that:

$\frac{\text{Area of shaded region}}{\text{Area of square}} = \frac{\pi}{8}$