KA-0052Area by subtraction (shaded regions)
5 points, difficulty 3 of 3Level 5–6about 135sA circle of radius 5 is drawn inside a square of side 10, touching all four sides. What fraction of the square is outside the circle, to the nearest tenth?
Text description of the figure
A square with a circle drawn inside it, the circle touching the midpoint of each of the four sides. The region inside the square but outside the circle is shaded.
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Find the whole and the hole, then subtract.
2The strategy
Work out the area of the square and the area of the circle separately, subtract to get the shaded region, and only then turn it into a fraction of the square.
3The full solution
The square is 10 x 10 = 100. The circle is pi times 5 squared, about 78.5. The region outside is about 21.5, which is about 0.2 of the square.
Solution
The reliable way
The square has area 10 x 10 = 100. The circle has radius 5, so its area is pi times 25, about 78.5. The shaded region is 100 - 78.5 = 21.5, which as a fraction of the square is 0.215, or 0.2 to the nearest tenth. The transferable idea: subtract to get the odd-shaped region, then convert to a fraction last.
The elegant way
The circle takes up pi over 4 of any square it is inscribed in, about 0.785, so the corners always keep about 0.215 - the side length never matters.
Why this is on the test: Inscribed-circle questions recur at every band, and the pi-over-four fact turns them into one step regardless of the numbers.