【Haichuan Teahouse】Brain-teasing Series: Big Circle Inside a Small Circle
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In the larger circle, there are two inscribed circles above and below it. The chord AB of the larger circle is the common tangent to these two inscribed circles, with the point of tangency being exactly the midpoint of AB. Given that AB = 8 cm, find the area of the shaded region.
;P The two circles form an 8 shape in between, so it’s 8π ;P;P;P
It’s really 8Π. If you take a close look, you can do it.
For 8Π, draw four auxiliary lines; then, using the formula for the area of a triangle, it can be shown that the diameter of the smallest circle is 1/4 of the diameter of the largest circle. Furthermore, applying the Pythagorean theorem, the diameter of the smaller circle can be determined to be 4*√3/3
Since there is no restriction on the size of the white circles, let’s consider a special case where the two white circles are of equal size. In this case, AB = 8 cm, which is the diameter of the outer circle; the diameter of each white circle is 4 cm. The area of the outer circle, S_outer, is 4×4×π = 16π, while the area of the inner circle, S_inner, is 2×2×π = 4π. Therefore, the area of the shaded part is S_out – 2S_white = 16Π – 2×4Π = 8Π
Based on your reasoning, I think it will be a maximum value with a minimum value
If the small circle is small enough, will it change?
This is the hidden condition of the problem. Since the size isn’t specified, it must be that the result is the same in all cases; if the results varied across different cases, then constraints would need to be provided.