TSA of Cone

Cone — Key Terms: • r = radius of circular base • h = perpendicular height (height from base to apex) • l = slant height (distance from apex to edge of base along the surface) • Relation: l² = r² + h² → l = √(r² + h²)

Formulae: • Curved Surface Area (CSA) / Lateral Surface Area (LSA) = πrl • Area of circular base = πr² • Total Surface Area (TSA) = CSA + Base Area TSA = πrl + πr² TSA = πr(l + r)

• Volume of cone = (1/3)πr²h

Derivation of CSA = πrl: • Unroll the curved surface of the cone → it forms a sector of a circle • Radius of sector = l (slant height); Arc length of sector = 2πr (circumference of base) • Area of sector = (1/2) × arc length × radius = (1/2) × 2πr × l = πrl

Solved Examples:

Example 1: Find TSA of a cone with r = 7 cm, l = 14 cm. CSA = πrl = (22/7) × 7 × 14 = 308 cm² Base = πr² = (22/7) × 49 = 154 cm² TSA = 308 + 154 = 462 cm²

Example 2: Find TSA of cone with r = 3 cm, h = 4 cm. l = √(r² + h²) = √(9 + 16) = √25 = 5 cm TSA = πr(l + r) = π × 3 × (5 + 3) = 24π = 75.43 cm²

Example 3: Find CSA of cone with r = 5 cm, l = 13 cm. CSA = πrl = π × 5 × 13 = 65π ≈ 204.2 cm²

Summary Table:

NCERT: Class 9 Maths Chapter 13 — Surface Areas and Volumes Value of π = 3.14159... or 22/7 (approx.)