Question 1: Calculate the length of the arc subtended by a 60° central angle in a circle with radius 10 cm.

Answer: 10π cm or approximately 31.42 cm

Question 2: Find the area of the segment cut off by a 90° central angle in a circle with radius 8 cm.

Answer: 8²/2 (π - 2) cm² or approximately 12.57 cm²

Question 3: A circle has a radius of 15 cm. A segment subtends a 45° angle at the centre. Calculate the area of the segment.

Answer: Approximate: 23.56 cm² (using segment area formula)

Question 4: Construct a circle with radius 12 cm and draw a segment that subtends a 120° central angle. Label the key parts.

Answer: Student's construction

Question 5: A circle with radius 9 cm has a segment whose area is 35 cm². Determine the size of the central angle that subtends this segment.

Answer: Approximately 70°

Question 6: In a circle with radius R, the segment area is given by the formula: Segment Area = (θ/360) * πR² - (1/2) R² sin θ. Explain why this formula is valid for calculating the area of a segment.

Answer: It subtracts the triangular area from the sector area to find the segment area.

Question 7: A sector with a radius of 20 cm has an angle of 150°. Calculate the area of the segment formed when this sector is cut off by a chord across the circle.

Answer: Approximately 69.4 cm²

Question 8: A student makes a common mistake by using the full sector area formula without subtracting the triangular part. Identify the error in calculating the segment area and suggest the correction.

Answer: They calculate the sector area but forget to subtract the triangular segment, leading to an overestimate.

Question 9: Given a circle with radius 6 cm, a segment subtends a 30° angle. Calculate the length of the chord that forms this segment.

Answer: Approximately 3.1 cm (using chord formula)

Question 10: A circle has a radius of 18 cm. The segment subtends a 120° angle. Calculate both the length of the chord and the area of the segment.

Answer: Chord length: approximately 18.62 cm; Segment area: approximately 60.4 cm²

Question 11: Explain the significance of the sine function in calculating the area of a segment and demonstrate with a relevant example.

Answer: The sine function relates to the height of the triangle formed, essential in calculating the segment's area when angles and radius are known.