I'm glad you asked that mak .... I'll have to walk myself through to make sure I get it myself ......

Here's a diagram which will suit our purpose.... It relates to the pressure in a tube and the stresses that produces , but the calculation is the same for our situation.

the diagram on the left shows the thread , all the arrows are the forces from the surface tension ... to find the tension in the thread we have to add all the components of the small forces together.

In the diagram on the right , T is the tension we need to find . The 2 small arrows pointing up and down will not impart any force because they are at right angles ... the small arrow pointing right will transmit all of it's force ... and all the arrows in between just some of their force ... you have to multiply by cos the angle to find out how much each arrow contributes .....

And when you add all these little forces together by integration it comes out to be 1/2pi