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Photo: DER SPIEGEL
The past week was about calculating with x and y.
Now follows a classic geometry knack from my new book "Blind Date with Two Unknowns".
There are six circles of equal size.
They are placed next to each other so that each circle touches two neighboring circles.
In addition, their centers form a regular hexagon - see drawing above.
How big is the area enclosed by the circles, the one in the drawing dark red
is colored?
Note:
The radius of the six circles should be 1.
The area is
6 * root (3) - 2 * Pi = 4.11.
If we connect the centers of the six circles, we get a regular hexagon.
The area enclosed by the circles then corresponds exactly to the area of the hexagon, from which we still have to subtract the six sectors of the circle - see the following drawing:
The area of the hexagon corresponds to six times the area of a regular triangle with edge length 2. Such a triangle has a height of root (3), which can easily be calculated using the Pythagorean theorem (length hypotenuse = 2, length short cathetus = 1, Height = Root (2 * 2 - 1 * 1) = Root (3)).
The triangular area is therefore 2 * root (3) / 2 = root (3).
For the hexagon we therefore get 6 * root (3) as the area.
Each of the six sectors of the circle corresponds to a third of the area of the circle - in total we therefore have to subtract twice the area of the circle - i.e. 2 * Pi.
For the enclosed area we therefore get: 6 * root (3) - 2 * pi
If you missed a puzzle from the past few weeks, here are the ten most recent episodes:
x, y unsolved
Floor plan with gaps
Wrong track
Seven in one go
Hundred square numbers
On the brink
Secret access code
Who lies?
Who is telling the truth
How does the sequence of numbers continue?
The round is in the square
