Question

could a circular pie be cut such that all of the wedge shaped pieces have an arc length along the outer crust equal to the pie's radius?

Answer 1

Circumference = pi*diameter OR 2*pi*r, therefore the ratio of circumference:radius is 2PI. Maybe if you had a circle with circumference of multiples of pi, say 8 x 3.141... = 25.128, but pi is a transcendental number so no, probably not

Answer 2

No -- as stated, in order for each wedge to have an arc length of the radius of the pie, the pie would have to be cut into a non integer (specifically 2 pi ) number of pieces.

Answer 3

technically yes
The number of degrees requred so that the length of the arc =r is called 1 radien.
But since they are not whole numbers you can only get reallly close to r

Answer 4

^^ notice it says "all the pieces". You'll have 6 really close pieces and then ~0.28.... of a piece left over, so the answer is unequivocally no.