Arc Length, Sector Area and Segments

A pizza cut from the centre by equal angles gives equal slices. An arc is a fraction of the circle's edge. Divide the central angle by 360 for the fraction, then multiply by the circumference, 2 pi r. The radius is half the diameter, so circumference is also pi times diameter. A sector is bounded by two radii and an arc. A segment by a chord and an arc. Keep pi as a symbol when asked, or use 3.14. A 90 degree angle gives a quarter. To find radius from an arc and nonzero angle, multiply arc by 360 over the angle, then divide by 2 pi.

A minor segment is what is left when you cut the triangle out of its sector, for a central angle between 0 and 180 degrees. Work out the sector, work out the triangle, then subtract. The triangle has two sides equal to the radius and the central angle between them, so its area is one half times r squared times sin of the angle. For a 90 degree angle this is one half r squared, the right-angled case. At 180 degrees the triangle vanishes and the segment equals a semicircle.

For a major segment, with the angle between 180 and 360 degrees, the segment is the sector plus the triangle, since the triangle area is positive.

A 180 degree sector is exactly half the circle, for arc length and area alike. A 360 degree sector is the whole circle. Use these two facts to sense check every answer.