What is the Curved Surface Area (CSA) of a Cone?

The Curved Surface Area of a cone is given by:

CSA = πrl

Where:

• π (pi) ≈ 3.14159
• r = radius of the circular base
• l = slant height of the cone

The slant height l is not the vertical height. It is the distance from the apex (tip) of the cone to any point on the circumference of the base. If the vertical height is h, then by the Pythagorean theorem:

l = √(r² + h²)

So the formula can also be written as: CSA = πr√(r² + h²)

A cone can be unrolled into a flat sector of a circle (like a slice of pizza). When you cut along the slant height and flatten it out, you get a sector with:

• Radius equal to the slant height l
• Arc length equal to the circumference of the base circle = 2πr

The area of a sector = (arc length / full circumference) × area of full circle = (2πr / 2πl) × πl² = (r / l) × πl² = πrl

Therefore, CSA of cone = πrl. This geometric unrolling approach gives us an intuitive and exact derivation of the formula.

There are two surface area measures for a cone:

1. Curved Surface Area (CSA) = πrl This covers only the slanted lateral surface, not the base.

2. Total Surface Area (TSA) = πrl + πr² This covers both the curved surface AND the circular base. TSA = πr(l + r)

Use CSA when the base is open (like an ice cream cone). Use TSA when the base is closed (like a party hat that is sealed at the bottom).

Example 1: Find the CSA of a cone with radius = 7 cm and slant height = 10 cm. CSA = πrl = π × 7 × 10 = 70π ≈ 219.91 cm²

Example 2: Find the CSA of a cone with radius = 6 cm and vertical height = 8 cm. First find slant height: l = √(r² + h²) = √(6² + 8²) = √(36 + 64) = √100 = 10 cm CSA = πrl = π × 6 × 10 = 60π ≈ 188.49 cm²

Example 3: A cone has CSA = 550 cm² and radius = 7 cm. Find its slant height. CSA = πrl → 550 = (22/7) × 7 × l → 550 = 22l → l = 25 cm

Example 4: Find the CSA of a cone with diameter = 14 cm and height = 24 cm. r = 14/2 = 7 cm; l = √(7² + 24²) = √(49 + 576) = √625 = 25 cm CSA = π × 7 × 25 = 175π ≈ 549.78 cm²

Important points to remember:

• CSA is always expressed in square units (cm², m², etc.).
• Never confuse height (h) with slant height (l). Always compute l = √(r² + h²) when only h is given.
• If diameter is given, divide by 2 to get radius before applying the formula.
• When using π = 22/7, it works cleanly when r or l is a multiple of 7.
• For exact answers, leave the result in terms of π (e.g., 70π cm²). For approximate answers, multiply by 3.14 or 3.14159.