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AQA GCSE · Question 12 · Geometry and Measures
A sector has radius 12 cm and angle 60°. Work out the length of the arc. Give your answer in terms of π.
A sector has radius 12 cm and angle 60°. Work out the length of the arc. Give your answer in terms of π.
How to approach this question
1. The formula for arc length is (θ/360) × 2πr, where θ is the angle and r is the radius.
2. Identify the values for θ (60°) and r (12 cm) from the question.
3. Substitute these values into the formula.
4. Simplify the fraction (60/360).
5. Calculate the final value, leaving π in your answer as requested.
Full Answer
Step 1: Write down the formula for the circumference of a full circle: C = 2πr.
Step 2: Calculate the circumference of the full circle with radius 12 cm.
C = 2 × π × 12 = 24π cm.
Step 3: The arc is a fraction of the full circumference. The fraction is the angle of the sector divided by the total angle in a circle (360°).
Fraction = 60° / 360° = 6/36 = 1/6.
Step 4: Multiply the full circumference by this fraction to find the arc length.
Arc Length = (1/6) × 24π = 24π / 6 = 4π cm.
Answer: 4π cm
The length of an arc of a sector is a fraction of the circumference of the full circle from which the sector is taken.
1. **Find the fraction of the circle:** The angle of the sector is 60° and a full circle is 360°. So the fraction is 60/360. This simplifies to 6/36 = 1/6.
2. **Find the circumference of the full circle:** The formula for circumference is C = 2πr or C = πd. The radius `r` is 12 cm.
C = 2 × π × 12 = 24π cm.
3. **Calculate the arc length:** Multiply the full circumference by the fraction.
Arc Length = (1/6) of 24π
Arc Length = (1/6) × 24π = 4π cm.
Common mistakes
✗ Using the formula for the area of a sector (θ/360 × πr²) instead of arc length.
✗ Using the diameter instead of the radius, or vice-versa.
✗ Making a mistake when simplifying the fraction 60/360.
✗ Multiplying by π instead of leaving it in the answer.
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