Very Difficult Problem For Many Students – Mind Your Decisions

A square has side length equal to 1 meter. A circle is constructed through one corner of the square and tangent to the opposite two sides of the square. What is the area shaded in blue below, which is the area contained by the square and circle excluding the region where the two shapes overlap?

As usual, watch the video for a solution.

Very Difficult Problem For Many Students

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Answer To Very Difficult Problem For Many Students

(Pretty much all posts are transcribed quickly after I make the videos for them–please let me know if there are any typos/errors and I will correct them, thanks).

The overlapping region can be partitioned into a triangle and a circular sector. Since the triangle is a right triangle, the inscribed angle is a 90 degree angle and therefore subtends a 180 degree arc. Therefore the hypotenuse of the triangle is a diameter of the circle and the circular sector is a semicircle. We thus have:

shaded area
= circle + square – 2(overlapping)
= circle + square – 2(semicircle + triangle)

But note 2 times the semicircle’s area is precisely the circle’s area, so those terms cancel out. We thus have:

shaded area
= square – 2(triangle)

It remains to calculate the area of the triangle. Let r be the circle’s radius. Construct radii to the right angle of the triangle, and to the two tangency points between the circle and square. The right triangle is then divided into 2 congruent isosceles right triangles with legs equal to r, so the hypotenuse of each is r√2. The upper two radii form a square with the upper part of the 1 meter square, so the small square’s diagonal is r√2.

The diagonal of the 1 meter square is √2, but it is also equal to r + r√2. Thus we have:

r + r√2 = 1
r(1 + √2) = 1
r = 1/(1 + √2)

Multiplying the numerator and denominator by the conjugate (1 – √2) and then simplifying gives:

r = 2 – √2

We can then calculate the shaded area:

shaded area
= square + 2(triangle)
= 1² + 2(r√2 · r√2)/2
= 1 + 2r²
= 1 + 2(2 – √2)²
= 8√2 – 11 square meters
≈ 0.314 square meters