What is the surface area of a sphere with a radius of 14√5 cm (use π = 22/7)?

Concept: The surface area of a sphere is fixed by its radius alone, through the identity S = 4πr2. The radius enters only as a square, so a radius written as a surd is squared exactly using (a√b)2 = a2b, and no decimal approximation of the surd is needed at any stage.

Application: here the radius is 14√5 cm and the question fixes π = 22/7.

1. Square the radius exactly: r2 = (14√5)2 = 142 × 5 = 196 × 5 = 980 cm2.

2. Substitute into the identity: S = 4 × (22/7) × 980 cm2.

3. Cancel the 7 against 980 before multiplying: 980 ÷ 7 = 140, which leaves S = 4 × 22 × 140.

4. Multiply the remaining factors: 4 × 22 = 88, and 88 × 140 = 12320.

5. Therefore S = 12320 cm2.

Cross-check: reverse the arithmetic. 12320 ÷ 4 = 3080, and 3080 × 7/22 = 980, which is exactly r2; since √980 = √(196 × 5) = 14√5, the radius recovered is the one the question gave, so the value is consistent.

Contrast: the factor in front of πr2 decides which figure is being measured — 2πr2 is the curved surface of a hemisphere (6160 cm2 here) and 3πr2 is the total surface of a hemisphere (9240 cm2 here). A full sphere always carries the factor 4.