Find the length of the arc of a circle whose central angle is 45° and whose radius is 28 cm, giving your answer in centimetres using the standard arc length formula.

Introduction / Context:
This geometry question uses the standard formula for the length of an arc of a circle. Given the central angle in degrees and the radius, you need to compute the arc length, which is a portion of the full circumference. Such problems are very common in school mathematics and quantitative exams and help reinforce the relationship between degrees, circumference, and proportional reasoning.

Given Data / Assumptions:

• The radius of the circle is r = 28 cm.
• The central angle is θ = 45°.
• We are asked to find the length of the corresponding arc.
• The entire circle corresponds to 360° around the centre.

Concept / Approach:
The length of an arc corresponding to a central angle θ (in degrees) is given by the fraction θ/360 of the entire circumference. The full circumference of a circle is 2πr. Therefore, the arc length s can be written as s = (θ/360) × 2πr. Once we substitute θ and r into this formula, the calculation becomes straightforward, and we can either leave the answer in terms of π or approximate it to a nearby integer value if the options are given in that form.

Step-by-Step Solution:
Step 1: Recall the arc length formula for degrees: s = (θ/360) × 2πr.Step 2: Substitute θ = 45° and r = 28 cm into the formula.Step 3: Compute the fractional factor: θ/360 = 45/360 = 1/8.Step 4: Compute the full circumference: 2πr = 2π × 28 = 56π cm.Step 5: Now calculate s = (1/8) × 56π = 7π cm.Step 6: If we approximate π as 22/7, then 7π ≈ 7 × 22/7 = 22 cm.

Verification / Alternative check:
Since 45° is one eighth of a full revolution (360°/8 = 45°), the arc length corresponding to 45° must be one eighth of the full circumference. The full circumference for radius 28 cm is 56π cm, so one eighth of that is 7π cm, confirming our earlier computation. Using π ≈ 3.14 also gives 7 × 3.14 ≈ 21.98 cm, which is very close to 22 cm and aligns with the nearest centimetre value in the options.

Why Other Options Are Wrong:
The value 11 cm is about half of the correct approximate value and would correspond to a much smaller central angle. The values 33 cm and 44 cm are too large; 44 cm is approximately two full radii and would require a larger sector. The exact value 7π cm is mathematically correct, but if the options use integer approximations, the matching choice is 22 cm. Among the numerical options provided, only 22 cm matches the correct approximate arc length.

Common Pitfalls:
Students sometimes mistakenly use θ/180 instead of θ/360 when working in degrees, which doubles the arc length. Others forget to multiply by 2πr and instead use πr, which is half the circumference. Keeping the correct formula in mind and recognising that 45° is one eighth of a full circle helps prevent such errors and leads quickly to the correct result.

Final Answer:
The length of the arc for a 45° central angle in a circle of radius 28 cm is 22 cm (approximately, corresponding to the exact value 7π cm).