Hello, Ilsa!

Code:

V
*
/|\
/ | \
/ | \
C * __| \ 6
/ √35| \
/ | \
/ | \
A * - - - * - - - * B
O 1

We have a right triangle with legs 1 and

The hypotenuse (slant height of the cone) is 6.

If we "unroll" *half* the cone and press it flat,

. . we have the diagram below.

Code:

V
*
/ \
/30°\ 3
/ \ C
6 / *
/ * \
/ * \ 3
/ * \
B * - - - - - - - *
* * *
π

The circumference of the base of the cone is: .

Half the circumference is:

We have: .

. . Hence: .

The vertex angle is 30°.

The shortest distance between and is a straight line.

In , Law of Cosines:

. .

Therefore: .