Henry Ernest Dudeney/Modern Puzzles/116 - The Crescent and the Star/Solution
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Modern Puzzles by Henry Ernest Dudeney: $116$
- The Crescent and the Star
- Here is a little puzzle on the Crescent and the Star.
- Look at the illustration, and see if you can determine which is the larger, the Crescent or the Star.
- If both were cut out of a sheet of solid gold, which would be more valuable?
- As it is very difficult to guess by the eye,
- I will state that the outer arc is a semicircle;
- the radius of the inner arc is equal to the straight line $BC$;
- the distance in a straight line from $A$ to $B$ is $12$ units,
- and the point of the star, $D$, contains $3$ square units.
Solution
They are both the same area: $36$ square units.
Proof
There are $12$ instances of the small equilateral triangle that forms the point $D$.
Hence the area of the star is $12 \times 3 = 36$ square units.
From Lune of Hippocrates, the area of the crescent is equal to the area of the square whose diagonal equals $BC$.
The side of this square is half the distance $AB$, which is $6$.
Hence from Area of Square, the area of the crescent is equal to $6 \times 6$, that is, $36$ square units.
$\blacksquare$
Sources
- 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Solutions: $116$. -- The Crescent and the Star
- 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $339$. The Crescent and the Star