This shape consists of three semicircles - OCR - GCSE Maths - Question 21 - 2018 - Paper 1

The area of a semicircle is given by the formula:

$ext{Area} = \frac{\pi r^2}{2}$

For the smaller semicircles A and B, the area can be calculated as:

$\text{Area A} = \frac{\pi (x)^2}{2} = \frac{\pi x^2}{2}$ $\text{Area B} = \frac{\pi (x)^2}{2} = \frac{\pi x^2}{2}$

Thus, the combined area of A and B is:

$\text{Area A + B} = \frac{\pi x^2}{2} + \frac{\pi x^2}{2} = \pi x^2$

Using the radius we found earlier, the area of the larger semicircle C is:

$\text{Area C} = \frac{\pi (2x)^2}{2} = \frac{\pi \cdot 4x^2}{2} = 2\pi x^2$

Now, we combine the areas of the three semicircles:

$\text{Total Area} = \text{Area A + B + Area C} = \pi x^2 + 2\pi x^2 = 3\pi x^2$

Thus, we have shown that the area of the whole shape is indeed 3πx².