Henry Ernest Dudeney/Modern Puzzles/116 - The Crescent and the Star/Solution

Modern Puzzles by Henry Ernest Dudeney: $116$

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.

They are both the same area: $36$ square units.

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.

$\blacksquare$

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