In a circle what name is given to a part of the circle bounded by a chord and an arc?
Area of a circle the excess amount. Inscribe a square in the circle, so that its four corners lie on the circle. Between the square and the circle are four segments. If the total area of those gaps, "G", is greater than "E", split each arc in half. This makes the inscribed square into an inscribed octagon, and produces eight segments with a smaller total gap, "G". Continue splitting until the total gap area, "G", is less than "E". Now the area of the inscribed polygon, "P" = "C" − "G", must be greater than that of the triangle. But this forces a contradiction,
Area of a circle on a diameter. By Thales' theorem, this is a right triangle with right angle at B. Let the length of A′B be "c", which we call the complement of "s"; thus "c"+"s" = (2"r"). Let C bisect the arc from A to B, and let C′ be the point opposite C on the circle. Thus the length of CA is "s", the length of C′A is "c", and C′CA is itself a right triangle on diameter C′C. Because C bisects the arc from A to B, C′C perpendicularly bisects the chord from A to B, say at P. Triangle C′AP