What is the total surface area of a cone whose curved surface area is 550 cm² and radius is 7 cm? (Take π = 22⁄7)

Correct — A. Curved surface area of a cone is πrl, so the slant height has to come first.

πrl = 550
(22⁄7) × 7 × l = 550
22l = 550, so l = 25 cm

Total surface area = πrl + πr²
= 550 + (22⁄7) × 7 × 7
= 550 + 154
= 704 → option (a).

Why the others are wrong

• (b)
  714.05 cm — 714.05 implies a base circle of 164.05 cm². With r = 7 and π = 22⁄7 the base is exactly 154 cm², and the stem's π leaves no decimal tail to carry.
• (c)
  682.60 cm — 682.60 implies a base of 132.60 cm², about what a 6.5 cm radius would give. The radius is fixed at 7 cm throughout.
• (d)
  764.71 cm — 764.71 implies a base of 214.71 cm², which needs a radius near 8.3 cm. Nothing in the stem changes r between the curved surface and the total.

A cone has two surfaces: the curved one, πrl, wrapped around the slant height, and the flat base, πr². Total surface area is their sum, πr(l + r).

Here the curved surface is given, so you work backwards to l before adding the base. The slant height l runs from the apex down the surface to the rim and is not the vertical height h, the two being linked by l² = r² + h².

Key facts

• Curved surface area of a cone is πrl, where l is the slant height rather than the vertical height.
• Total surface area of a cone is πrl + πr², which factorises to πr(l + r).
• With π = 22⁄7 and r = 7 the product πr is exactly 22, so πrl reduces to 22l.
• The cone here has l = 25 cm, so its vertical height is √(25² − 7²) = 24 cm.

Common traps

• Stopping at 550 and reporting the curved surface as the total.
• Putting the vertical height 24 cm into πrl instead of the slant height 25 cm.
• Reading the printed cm as a length and hunting for a perimeter.

No directly related past PYQ was found.