Semi-Circular Logic
A green simicirle is inscribed inside a right triangle as shown. What is the area of the semicircle?
Brainwasher
Find the areas of the two annuli (green shaded “washers”) in the figure below (hint: they have the same area).
Checkmate
Can you cut the pattern shown into exactly 2 pieces and reassemble them to make a standard 8-by-8 chessboard?
See Answers Below!
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Semi-Circular Logic
Let’s show r as the radius of the semicircle and derive one additional dimension:
The small triangle in the upper left is also a right triangle and is “similar” to the larger overall triangle, so we know that the ratio of their sides is equal (both of these values are equal to the tangent of the upper angle of the triangle – we can state the equality without knowing that exact angle):
And solve for r
With the radius(r) known, we plug it into the equation for the area of a semicircle :
Brainwasher
Let’s label the radius of the smaller circle as r, and the larger circle as R, and note that half the chord is length 6. And note that these three lines form a right triangle.
So we can write two equations, the first is the relationship between r and R:
The second is Area(A) of the green shaded portion:
Then substitute from the first into the second:
The exact same dimensions, variables, and triangle can be drawn/applied to the other figure, and the math is identical. The size of the circles can vary infinitely. As long as a chord (of length 12) spans the larger circle and is tangent to the smaller, then the area of the “washer” will always be 36π.
(It’s also interesting to consider the extreme case where the radius of the inner circle is reduced to 0. The green-shaded area is then simply the area of a circle with a radius of 6.)
Checkmate
Cut the figure as shown:
Then rotate the inner part 90 degrees clockwise, and overlay:
