Question 1: What is the fraction of the circumference when the arc length measures 6 cm and the full circumference is 24 cm? Show your calculation.

Answer: 1/4

Question 2: Calculate the arc length of a sector if the circle's radius is 7 cm and the sector represents 3/8 of the full circumference.

Answer: 16.5 cm

Question 3: A student incorrectly states that the arc length for 1/2 of a circle is 12 cm when the diameter is 10 cm. Identify the mistake and correct it.

Answer: The diameter is 10 cm, so the circumference is π×10 ≈ 31.4 cm. Half of that is about 15.7 cm, not 12 cm.

Question 4: Construct a circle on the grid with a radius of 4 units. Mark an arc that is 1/3 of the total circumference. Show the arc on the grid.

Answer: Student draws quarter circle and marks 4π/3 units on the arc.

Question 5: A circle has a radius of 9 cm. What is the length of an arc that measures 1/6 of the circle's circumference? Provide your answer with calculation.

Answer: ABOUT 9.42 cm

Question 6: A student mistakenly believes that 1/5 of the circumference is 8 cm when the radius is 5 cm. Explain the error and find the correct arc length for 1/5 of the circle.

Answer: Incorrect because circumference = 2π×5 ≈ 31.4 cm. 1/5 of that is about 6.28 cm, not 8 cm.

Question 7: Express the length of an arc that is 2/3 of a circle with radius 10 cm as a numerical value.

Answer: ≈ 20.94 cm

Question 8: Identify the common misconception in calculating arc length as simply multiplying the fraction by the radius instead of the full circumference. Correct the mistake with an example.

Answer: Incorrect method: arc length = fraction × radius. Correct method: arc length = fraction × circumference = fraction × 2π×radius.

Question 9: A circular track has a radius of 15 meters. What is the length of the arc corresponding to 1/4 of the track? Show your work.

Answer: ≈ 23.56 meters

Question 10: A student calculates that 1/3 of the circumference is 10 cm when the radius is 5 cm. Explain whether this is correct and justify your answer.

Answer: Incorrect because circumference = 2π×5 ≈ 31.4 cm. 1/3 of that is about 10.47 cm, so 10 cm is slightly underestimated.