Competition problem — Geometry — Maths Olympiad Prep

Library / /105 of 520

Geometry Difficulty 2.1 Junior Find the answer

A circle of diameter $1$ is removed from a $2\times 3$ rectangle, as shown. Which whole number is closest to the area of the shaded region?
[asy] fill((0,0)--(0,2)--(3,2)--(3,0)--cycle,gray); draw((0,0)--(0,2)--(3,2)--(3,0)--cycle,linewidth(1)); fill(circle((1,5/4),1/2),white); draw(circle((1,5/4),1/2),linewidth(1)); [/asy]

Pick one

Solution

The area of the shaded region is the area of the circle subtracted from the area of the rectangle.
The diameter of the circle is $1$ , so the radius is $1/2$ and the area is
$(1/2)^2\pi = \pi /4.$
The rectangle obviously has area $2\times 3= 6$ , so the area of the shaded region is $6-\pi / 4$ .
This is closest to $5\rightarrow \boxed{\text{E}}$ .

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.