The dots around the semicircle are equally spaced. What fraction is shaded?
This puzzle is by the amazing Catriona Agg.
This becomes very easy once we add an auxiliary
regular hexagon with its diagonals.
We will show that the largest of the three shaded regions (blue in figure a) is exactly
one third
of the total area
This is obvious because the triangles in b and c have the same area.
Because they have
or some may prefer because
or others still may skip c completely
Easy.
Next we'll show that the blue shaded area and the non-shaded area are
the same
and we'll be done.
First, we move the tiny shaded bit (Figure d) and then we are left with a matching curved bit (Figure e) and again two triangles that share their base and have the same height, half the radius (Figure f).
To summarise
The blue, teal and non shaded regions split the total area evenly. In particular, 2/3 are shaded.
My answer is
Two-thirds of the semi-circle is shaded
The area of a sector is (area of the segment) - (area of triangle subtended)
Suppose the angle subtended is $\theta$, the segment area
$= (\pi r^2 \times \frac{\theta}{360}) - (r \sin \frac{\theta}{2} \times r \cos \frac{\theta}{2}) $
$= r^2 (\frac{\theta \pi}{360} - \frac{\sin \theta}{2})$
Each dot is spaced at 30°, so
Area of a 30° segment
A1 = $ r^2 ( \frac{30 \pi}{360} - \frac{\sin 30}{2})$
Area of a 60° segment
A2 = $ r^2 ( \frac{60 \pi}{360} - \frac{\sin 60}{2})$
Area of a 120° segment
A4 = $ r^2 ( \frac{120 \pi}{360} - \frac{\sin 120}{2})$
Area of a 150° segment
A5 = $ r^2 ( \frac{150 \pi}{360} - \frac{\sin 150}{2})$
Area of a 180° segment
A6 = $ r^2 ( \frac{180 \pi}{360} - \frac{\sin 180}{2})$
We want the area
A1 + (A4 - A2) + (A6 - A5)
Which is
$r^2( \frac{(30 + 120 - 60 + 180 - 150) \pi}{360} - \frac{(\sin 30 + \sin 120 - \sin 60 + \sin 180 - \sin 150)}{2})$
reducing to
$r^2( \frac{120 \pi}{360} - \frac{\sin 180}{2})$
and
$\frac{\pi r^2}{3}$
which is one third of a full circle.