Masha's dog Mitzi is tied to a corner of a walled garden with the shape of a regular hexagon 8 meters on a side. Mitzi's rope is 10 meters long and Mitzi is tied up outside the garden---usually. What is the area within which Mitzi can roam?
Answer the same question if Mitzi is tied up inside the garden.
Solution
Take a look at the figure below, particularly part (1). When Mitzi is tied outside the garden, she roams over the area ABDFJGK, which we abbreviate I. When tied inside the garden, the relevant area is ABECHGK, which we call II.
Area I consists of 2/3 of a circle with radius 10 and 2/6 of a circle of radius 2. Therefore, area I is
I=32π102+62π22=68π≈213.6
This is the easy one.
Area II consists of two triangles and a sector of a circle of radius 10. We need the angles x,y and z shown in part (2) of the figure. Remember: the vertex angle of a regular hexagon is 120°.
Now,
sin(120°)10=sin(x)8⇝x=arcsin(108sin(120°))≈43.9°
We next find that,
yz≈180−120−43.9=16.1,≈120−2(16.1)=87.8.
We can now find the areas relevant to II:
area triangle AEC area sector ACK≈218⋅10⋅sin16.1≈11.1≈36087.8π102=76.6.
Therefore the internal roaming area for Mitzi is
II≈76.6+2⋅11.1=98.8