The radius of the outer circle is the same as the diameter of of one of the inner circles

Using the formula, $A = π r^{2}$ , work out the area of the outer circle

$A_{o u t e r} = π \times 4^{2}$

$A_{o u t e r} = 16 π$

[1]

Find the radius of one of the inner circles

$r_{i n n e r} = 4 \div 2 = 2$

Calculate the area of one of the inner circles

$A_{i n n e r} = π \times 2^{2}$

$A_{i n n e r} = 4 π$

Double the result to find the total area of the two inner circles

$2 \times 4 π = 8 π$

Find the shaded area by subtracting the area of the two inner circles from the area of the outer circle

$\begin{array}{rcl} A_{s h a d e d} & = & A_{o u t e r} - A_{i n n e r} \end{array}$

$\begin{array}{rcl} A_{s h a d e d} & = & 16 π - 8 π \end{array}$

$A_{s h a d e d} = 8 π$

[1]

Divide the shaded area by the area of the outer circle to find the fraction of the diagram that is shaded

$\frac{A_{s h a d e d}}{A_{o u t e r}} = \frac{8 π}{16 π}$

[1]

Simplify the fraction by cancelling a factor of 8 and a factor of $π$

$= \frac{1}{2}$

Yes, Frances is correct, the shaded area is exactly $\frac{1}{2}$ of the total area of the diagram [1]

Exact Values (Edexcel GCSE Maths: Foundation)

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