The area of a circular sector is 462 square centimetres. If the radius of the circle is 21 cm, what is the measure, in degrees, of the central angle of this sector?

Introduction / Context:
This question involves the area of a sector of a circle, which depends on both the radius and the central angle. You are required to invert the usual area formula to find the angle when the area and radius are known. Such problems strengthen your understanding of proportional relationships between the angle of a sector and the full circle.

Given Data / Assumptions:

• Area of the sector = 462 square centimetres.

• Radius of the circle r = 21 cm.

• Let the central angle of the sector be theta degrees.

• Use pi = 22 / 7.

Concept / Approach:
The area of a full circle is pi * r^2. The area of a sector with central angle theta (in degrees) is given by (theta / 360) * pi * r^2. Here, the area of the sector and the radius are known, so we substitute these values into the formula and solve for theta by simple algebraic manipulation. Careful handling of numbers and cancellation can make the calculation easy.

Step-by-Step Solution:
Sector area = (theta / 360) * pi * r^2. Given area = 462 square centimetres and r = 21 cm. So, 462 = (theta / 360) * pi * (21)^2. Compute r^2: 21^2 = 441. Substitute pi = 22 / 7, so 462 = (theta / 360) * (22 / 7) * 441. Simplify 441 / 7 = 63, so 462 = (theta / 360) * 22 * 63. Compute 22 * 63 = 1386. So, 462 = (theta / 360) * 1386. Rearrange: theta / 360 = 462 / 1386. Simplify 462 / 1386 = 1 / 3. Thus, theta / 360 = 1 / 3, so theta = 360 / 3 = 120 degrees.

Verification / Alternative check:
We can check by computing the sector area using theta = 120 degrees. The fraction of the circle covered by the sector is 120 / 360 = 1 / 3. The area of the full circle is pi * r^2 = (22 / 7) * 441. Since 441 / 7 = 63, this equals 22 * 63 = 1386 square centimetres. One third of 1386 is 1386 / 3 = 462 square centimetres, which matches the given area, confirming that theta = 120 degrees is correct.

Why Other Options Are Wrong:
If theta were 90 degrees, the sector area would be one fourth of the full circle, equal to 1386 / 4 = 346.5 square centimetres, not 462. For 60 degrees, the area would be one sixth of the circle, equal to 1386 / 6 = 231 square centimetres. For 30 degrees, the area would be even smaller at 1386 / 12 = 115.5 square centimetres. Only 120 degrees yields exactly 462 square centimetres.

Common Pitfalls:
Students sometimes confuse radians and degrees, but here the formula clearly uses degrees. Another common issue is making arithmetic mistakes while simplifying 462 / 1386 or using pi incorrectly. Some candidates also mistakenly use the formula for arc length instead of area. Writing down the correct formula and simplifying step by step helps avoid these errors.

Final Answer:
The measure of the central angle of the sector is 120°.