What is the area of a sector of a circle whose central angle is 90° and whose radius is 14 cm, using the standard formula for the area of a sector in square centimetres?

Introduction / Context:
This geometry question involves the area of a sector of a circle, a common topic in school mathematics and aptitude exams. Instead of the full area of a circle, we only want the area corresponding to a central angle of 90°, which represents a quarter of the full circle. The problem tests your ability to use the sector area formula correctly and substitute the given radius and angle.

Given Data / Assumptions:

• Radius r of the circle is 14 cm.
• Central angle θ of the sector is 90°.
• We assume π is used in the usual way, often approximated as 22/7 in aptitude problems.
• We are asked to find the area of the sector in square centimetres.

Concept / Approach:
The area of a full circle is πr^2. A sector with central angle θ degrees represents θ/360 of the full circle. Therefore, the area of a sector is (θ / 360) * πr^2. With θ = 90°, the sector is exactly one quarter of the full circle, so its area is (1/4) * πr^2. Substituting r = 14 cm and π = 22/7 (a typical exam approximation) gives a numerical value.

Step-by-Step Solution:
Use the formula for sector area: Area = (θ / 360) * π * r^2. Here θ = 90° and r = 14 cm. So Area = (90 / 360) * π * 14^2. Simplify 90 / 360 = 1/4. Therefore, Area = (1/4) * π * 14^2. Compute 14^2 = 196. So Area = (1/4) * π * 196. In aptitude questions, take π = 22/7. Then Area = (1/4) * (22/7) * 196. First simplify (22/7) * 196: 196 / 7 = 28, and 28 * 22 = 616. So Area = (1/4) * 616 = 616 / 4. Finally, 616 / 4 = 154. Thus, the area of the sector is 154 square centimetres.

Verification / Alternative check:
We can also find the full area of the circle and then take a quarter. Full circle area = πr^2 = (22/7) * 196 = 616 sq cm. Since 90° is a quarter of 360°, the sector area is 616 / 4 = 154 sq cm, matching our previous calculation. This confirms that our use of the sector formula and arithmetic are correct.

Why Other Options Are Wrong:
308 sq cm and 231 sq cm correspond to other fractional parts of the circle or incorrect arithmetic with π and r^2. 77 sq cm is half of 154, which would be the area of a 45° sector, not a 90° one. 100 sq cm does not correspond to any simple fraction of 616 and arises only from miscalculations in either squaring the radius or dividing by 4.

Common Pitfalls:
Typical mistakes include forgetting to square the radius, using the wrong fraction θ / 360 (for example, 90 / 180 instead of 90 / 360), or incorrectly simplifying (22/7) * 196. Some students also mistakenly compute the circumference formula instead of the area formula. Carefully applying the correct formula and simplifying step by step prevents these errors.

Final Answer:
The area of the sector is 154 sq cm.