The lengths of the minor and major arcs 54cm and 126cm respectively. Calculate the angle of the major sector

Analyze the question

The circle is split into a minor arc (54 cm) and a major arc (126 cm). We need the angle of the major sector.

Concept

The angle of a sector is proportional to its arc length:

Angle of sector / 360° = arc length / circumference

The circumference is the sum of both arcs.

Solution

1. Circumference = 54 + 126 = 180 cm
2. Angle of major sector = (126 / 180) × 360°
3. 126/180 = 7/10, so the angle = 7/10 × 360° = 252°

Check: The minor sector angle = (54/180) × 360° = 108°. Then 108° + 252° = 360° ✓

Answer: 252°, which is option B.

Tip: When you are given two arcs, add them to get the circumference. Then use the fraction arc/circumference × 360°. Always check that the minor and major angles add up to 360°. Here, 126/180 simplifies quickly to 7/10, which makes the calculation easy.