1. why do we usually have to use three points to name a major arc?

2. if you didn’t have access to a calculator or computer and all you had was pencil and paper, how would you figure out what 135° was in radians?

1. Two points are generally interpreted as naming the shorter arc of a circle. If the longer arc is intended, it must be identified differently some way. It is easier to add a third point and say "arc ACB" than to say "long arc AB".

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2. If you didn't already know that 135° = 3/4 of a half-circle, hence 3/4π radians, then you could figure it out from ...

  135° = 135°×(π/180°) radians = π × (135/180)

Reducing the fraction 135/180 without a calculator can proceed several ways. One is ...

  135/180 = 270/360 = 27/36 = (3·9)/(4·9) = 3/4

Now, you know that ...

  135° = 3π/4 radians

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In general, a calculator will give you a numerical answer, such as 2.3561945 radians. If you want a numerical answer and don't have a calculator, you can use the ratio 355/113 for π and then do the long division. Only 6 significant figures will be accurate using this ratio.

  (3/4)·(355/113) = 1065/452 ≈ 2.35619 . . . . radians