# Finding unequal radius of "sector

Hi first time poster here!

I'm working with a shape similar to a sector but having "radii" or sides of different lengths. Something similar to the following Firstly, what would you call such a shape since the radii are unequal?

Secondly, I know angles A (30º) and C (90º) and the length of the curved side 'b' (1º/110.6 km), but I need to find side 'a'. I've been trying to get around this using spherical trigonometric equation:

sinb = tanacotA

But I'm not convinced this is the right approach as only one side of my shape is spherical.

I've also tried using the formula for arc length (a = angle at centre/360 * 2*Pi*R) to find the radius but once again I'm not sure this is the right approach and I have hit somewhat of a wall.

• Is the arc $AC$ a circular arc?
– MPW
Oct 17, 2016 at 13:31
• Yes with length of 1º Oct 17, 2016 at 13:34
• What isthe wall that you hit? Oct 17, 2016 at 13:34
• I need to find the length of 'a' but I am not sure which is the right way to go about it, considering that I have a sector with unequal sides or 'radii' Oct 17, 2016 at 13:36

To form an angle of 30° with the tangent, $AB$ must be part of a chord subtended by a central angle of 60° (see diagram below), so that your picture cannot be accurate - or possibly you didn't explain exactly your problem.
In the case of my diagram, by the sine law you have $${d\over\sin 60°}={r\over\sin119°},$$ so you can find $d$ as a function of $r$ and subsequently $a=r-d$.
Radius $r$ is given because arc $b$ has a length of 110.6 km: $${r}={360\over2\pi}\cdot110.6\ \hbox{km}.$$ 