What is the measure of the central angle, in degrees, of a circular arc whose length is 11 cm, given that the radius of the circle of which it is a part is 14 cm?

Introduction / Context:
In this problem we work with the relationship between the length of an arc and the central angle that subtends it. Such questions are very common in aptitude tests because they connect basic circle geometry with unit conversion between radians and degrees, and they test whether you know the standard formula for arc length as well as how to manipulate it to find the unknown angle.

Given Data / Assumptions:

• The circle has radius r = 14 cm.
• The length of the arc is l = 11 cm.
• The central angle that subtends this arc at the centre of the circle is unknown and must be found in degrees.
• We assume a standard Euclidean circle and use the usual relationship between arc length, radius and angle in radians.

Concept / Approach:
The key formula for arc length of a circle is:
l = r * θ where l is the arc length, r is the radius, and θ is the central angle in radians. Once we find θ in radians, we then convert it to degrees using the relation:
degrees = θ * (180 / π) Because the options are all in degrees, our final step is this conversion, usually with π approximated as 22/7 in aptitude problems.

Step-by-Step Solution:
Step 1: Write the arc length formula: l = r * θ. Step 2: Substitute l = 11 cm and r = 14 cm. Step 3: 11 = 14 * θ. Step 4: Solve for θ in radians: θ = 11 / 14. Step 5: Convert θ to degrees using degrees = θ * (180 / π). Step 6: Use π = 22/7. Then degrees = (11 / 14) * (180 / (22 / 7)). Step 7: Simplify: degrees = (11 / 14) * 180 * 7 / 22. Step 8: Notice that 11 cancels with 22 and 7 cancels with 14, giving degrees = 45. Step 9: Therefore the measure of the central angle is 45°.

Verification / Alternative check:
We can quickly verify the arithmetic. Starting from (11 / 14) * 180 * 7 / 22, combine numerators and denominators. The factor 11 in the numerator cancels with 22 leaving 2 in the denominator. The factor 7 in the numerator cancels with 14 leaving 2 in the denominator. We then have 180 / (2 * 2) = 180 / 4 = 45 degrees. This check confirms that there was no mistake in simplification or cancellation and that the angle is exactly 45 degrees, which is a nice standard value.

Why Other Options Are Wrong:
60° would correspond to a larger arc because for a fixed radius, a larger angle produces a longer arc than 11 cm. 75° and 90° are even larger, which would make the arc length significantly more than 11 cm for radius 14 cm. The value 135° is much more than 90°, giving an arc that is clearly too long. Only 45° is consistent with the formula and the given numbers.

Common Pitfalls:
A frequent mistake is to try to use degrees directly in the formula l = r * θ without converting to radians or without using a correct conversion. Another common error is to invert the relationship and write θ = r / l instead of θ = l / r. Some students also forget to use π = 22/7 when simplifying, which makes the arithmetic look messy and can lead to rounding errors. Finally, mixing up the roles of radius and diameter in the formula can lead to an answer that is off by a factor of two.

Final Answer:
Thus, the measure of the required central angle is 45°.