Given two points A and B, lines from them to a third point P form the inscribed angle ∠APB. As you drag the point P above, notice that the inscribed angle is constant. It only depends on the position of A and B.
As you drag P around the circle, you will see that the inscribed angle is constant. But when P is in the minor arc (shortest arc between A and B), the angle is still constant, but is the supplement of the usual measure. That is, it is 180-m, where is m is the usual measure.
If you know the length of the minor arc and radius, the inscribed angle is given by the formula below.where:
The formula is correct for points in the major arc. If the point is in the minor arc, then the will produce the supplement of the correct result, but the the length of the minor arc should still be used in the formula.
Refer to the above figure. If the two points A,B form a diameter of the circle, the inscribed angle will be 90°, which is Thales' Theorem. You can verify this yourself by solving the formula above using an arc length of half the circumference of the circle.
You can also move the points A or B above until the inscribed angle is exactly 90°. You will see that the points A and B are then diametrically opposite each other.