Maximised Area
Of these five figures, which shaded area is the greatest? The large circle in each figure has the same radius.
Age
14 to 16
Challenge level
Image
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.
Answer:
Image
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:
Image
Image
Image
Area = $\pi\times1^2$ $ - $ $2\times\left(\pi\times\left(\frac12\right)^2\right)$
= $\pi$ $-$ $\frac12\pi$ = $\frac12\pi = 1.571$
B:
Image
Image
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:
Image
Image
Image
D:
Image
Image
E:
Image
Image
Image
= $\pi$ $-$ $\frac12(1\times1 )\times4 =\pi-2 = 1.142$