Geometry on a sphere?

Question

i was asked to find the formula for the circumfrance of a circle on a sphere. This formula's varibles can only be the radius of the sphere and the radius of the circle on the sphere (however this radius has to be the distance on the sphere i.e. the surface distance.) Please do not post any answers useing pie because that is not part of the formula

Answer 1

Let E be the raduis of the earth
Let r be the actual radius of the circle in the plane
Let S be the radius of the circle on the sphere
Let θ be the angular separation of the centre of the circle on the sphere to the circle in degrees

Thus θ = πθ/180 radians

Then C = 2πr
r = E*sinθ → C = 2πE*sinθ
Also S = E*πθ/180 → E = 180*S/(πθ)

Thus C = 2π*sinθ*180*S/(πθ)
ie C = 360Ssinθ/θ
where S = radius on sphere and θ is the angular separation (at the centre of the sphere) between the centre of the circle and any point on its circumference, expressed in degrees

Answer 2

A good question. But this does not make sense.

" (however this radius has to be the distance on the sphere i.e. the surface distance.) " This is not possible as the surface of the sphere cannot be a straight line, if I am understanding the question correctly.

It looks like to be an integration question with solid angle.

Regards,
Sandy

Answer 3

If you call 'R' the radius of the sphere, and 'r' the "surface radius" of the circle, then the circumference 'C' can be calculated with:

C= 2*pi*R*sen ((180° * r)/(pi*R))

You may simplify this expression further if you like.

Answer 4

i think you will have to take into account the formula "four thirds pie times the radius cubed".however it would seem impossible to produce a formula as the sphere changes width as it increases up in height.because the height varies it seems impossible to produce a formula

Answer 5

This is a guess
integration of function
rA wrt dA
within limits 0to A
where r is radius of sphere
and dA is the small angle
subtended by center of sphere
on arc of circle