Video narration
How can one circle and one chord produce four different answers? Think of a pizza slice: its wedge is like a sector, while the curved cap beside the crust is like a segment. One piece of geometry, four connected calculations. Here, circle center O has radius 10 centimeters, and chord AB is 16 centimeters long. Drop perpendicular OM from O to AB at M. Because a perpendicular from the center to a chord bisects it, AM equals MB, and each is 8 centimeters. OA and OB are radii, so each is 10 centimeters. Now use right triangle OMA. By Pythagoras, OM squared plus AM squared equals OA squared. So OM squared plus 8 squared equals 10 squared. OM squared equals 100 minus 64, which equals 36. Therefore, OM equals 6 centimeters. That same distance is the height of triangle AOB. Its area is one-half times 16 times 6, giving 48 square centimeters. For the angle, in triangle OMA, sin of angle AOM equals 8 over 10, or 0.8. Inverse sine gives angle AOM approximately 53.13 degrees. Since OM bisects the central angle, ∠AOB ≈ 106.26°. The minor sector is the region between OA, OB, and the minor arc. Its area is (106.26 over 360) times pi times 10 squared, approximately 92.73 square centimeters. The minor segment is only the curved cap between that arc and chord AB. So subtract the triangle: 92.73 minus 48 equals approximately 44.73 square centimeters. So how does one circle and chord give four answers? The distance is 6 centimeters, triangle area is 48 square centimeters, minor sector area is 92.73 square centimeters, and minor segment area is 44.73 square centimeters.