This question tests understanding of circle sectors. The area of a sector is (θ/360) × πr², where θ is the central angle. Given r = 9 and area = 27π/2, set (θ/360) × 81π = 27π/2, so θ/360 = (27/2)/81 = 1/6, θ = 60°. This solves for θ using the given area. This justifies that the central angle is 60°, which is choice C. A tempting incorrect option is 90°, perhaps from using r instead of r². Another error could be dividing by π incorrectly, leading to 30° or 120°.

This question tests knowledge of regular polygons. The interior angle is [(n-2) × 180°]/n. Setting this to 165° gives (n-2) × 180 = 165n, 180n - 360 = 165n, 15n = 360, n = 24. This solves the equation accurately. This justifies that the polygon has 24 sides, which is choice D. A tempting incorrect option is 18, perhaps from using exterior angle 15° incorrectly. Another common mistake is subtracting wrong, leading to 12 or 30.

This question tests understanding of circle sectors. The arc length is (θ/360) × 2πr. Given r = 6 and arc length 4π, (θ/360) × 12π = 4π, so θ/360 = 4/12 = 1/3, θ = 120°. This solves for θ using the arc formula. This justifies that the central angle is 120°, which is choice C. A tempting incorrect option is 90°, perhaps from using r instead of 2r in circumference. Another error could be using radians, leading to 60° or 240°.

This question tests the relationship between circumference and radius of a circle. The circumference formula is C = 2πr, where r is the radius. Given C = 18π, we solve for r: 18π = 2πr, which gives r = 18π/(2π) = 9. Therefore, the radius is 9 units. A common mistake is to confuse radius with diameter, which would incorrectly give 18 as the answer, or to divide by π instead of 2π.

This question tests the relationship between circumference and radius of a circle. The circumference formula is C = 2πr, where r is the radius. Given C = 18π, we solve: 18π = 2πr, which gives r = 18π/(2π) = 9. Therefore, the radius is 9. Choice A (18) is a common error where students confuse radius with diameter, forgetting that diameter = 2×radius.

This question tests knowledge of polygons and circles. In a regular hexagon inscribed in a circle, each side equals the radius. Given radius 10, each side is 10, so the perimeter is 6 × 10 = 60. This applies the property of equilateral sides in a regular hexagon. This justifies that the perimeter is 60, which is choice C. A tempting incorrect option is 50, perhaps from miscounting the sides as 5. Another common mistake is using the circumference instead, leading to values like 120.

This question tests the formula for the circumference of a circle. The circumference of a circle is given by C = 2πr, where r is the radius. With radius r = 7 centimeters, we calculate C = 2π(7) = 14π centimeters. Therefore, the circumference is 14π centimeters. A common mistake is confusing circumference with area, which would give πr² = 49π, leading to inCorrect answer B.

This question tests the area formula for a circle when given the diameter. The area of a circle is A = πr², where r is the radius. Given diameter d = 10 centimeters, the radius r = d/2 = 5 centimeters. Therefore, A = π(5)² = 25π square centimeters. The area is 25π square centimeters. A common mistake is using the diameter directly in the formula instead of the radius, which would incorrectly give 100π.

This question tests the formula for the sum of interior angles in a polygon. The sum of interior angles of any convex polygon with n sides is (n-2) × 180°. For an 11-sided polygon, this equals (11-2) × 180° = 9 × 180° = 1,620°. Therefore, the sum of interior angles is 1,620°. A common error is to miscalculate the multiplication or to use n instead of (n-2) in the formula, which would give 1,980°.

This question tests the perimeter formula for regular polygons. A regular octagon has 8 equal sides, and perimeter equals the number of sides times the length of each side. Given perimeter P = 72, we find the side length s by dividing: s = P/8 = 72/8 = 9. Therefore, each side has length 9 units. A common error is to confuse the number of sides (an octagon has 8 sides, not 6 or 12), leading to incorrect answers like 12 or 6.