# Maximised Area

Of these five figures, which shaded area is the greatest? The large circle in each figure has the same radius.

## Problem

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The large circles in each figure has the same radii.

Which shaded area is the greatest?

If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.

## Student Solutions

**Answer**:

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The large circles could have any radius and the answer will still be the same.

Choose a nice radius for the large circles, e.g. 1 unit

A:

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Area = $\pi\times1^2$ $ - $ $2\times\left(\pi\times\left(\frac12\right)^2\right)$

= $\pi$ $-$ $\frac12\pi$ = $\frac12\pi = 1.571$

B:

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Area of a triangle is $\frac12ab\sin{C} = \frac12\times1\times1\times\sin{120} = \frac{\sqrt3} 4 = 0.433$

Total area is $\frac{3\sqrt3}4 = 1.299$

C:

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D:

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E:

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= $\pi$ $-$ $\frac12(1\times1 )\times4 =\pi-2 = 1.142$