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

The area of each small semicircle (A and B) can be calculated using the formula for the area of a semicircle:

ext{Area of A} = ext{Area of B} = rac{1}{2} imes rac{ ext{π} imes r^2}{2} = rac{ ext{π} imes x^2}{2}.

Thus, the total area of semicircles A and B combined is:

$$ ext{Total Area (A + B)} = rac{ ext{π} imes x^2}{2} + rac{ ext{π} imes x^2}{2} = ext{π} x^2.$

The area of the large semicircle (C) can be calculated in a similar manner:

$$ ext{Area of C} = rac{1}{2} imes ext{π} imes (2x)^2 = rac{1}{2} imes ext{π} imes 4x^2 = 2 ext{π} x^2.$

To find the total area of the entire shape, we sum the areas of the small semicircles A and B and the large semicircle C:

$ext{Total Area} = ext{Area (A + B)} + ext{Area (C)} = ext{π} x^2 + 2 ext{π} x^2 = 3 ext{π} x^2.$

Thus, we have shown that the area of the whole shape is indeed:

$3 ext{π} x^2.$