01 · RECAP FORMULAS
Curved & Total Surface Areas of Basic Solids
Cylinder
Right Circular Cylinder
CSA = 2πrh, TSA = 2πr(h + r).Cone
Right Circular Cone
Slant Height l = √(r² + h²), CSA = πrl, TSA = πr(l + r).Sphere / Hemisphere
Sphere & Hemisphere
Sphere Area = 4πr². Hemisphere CSA = 2πr², Hemisphere TSA = 3πr².02 · GOLDEN RULE
Surface Area of Combined Shapes
CRITICAL RULE: DO NOT SIMPLY ADD TOTAL SURFACE AREAS!
When two solids are joined together, their touching base faces are hidden inside and no longer visible.
Total Surface Area of Combination = Sum of EXPOSED (VISIBLE) Curved Surface Areas.
COMBINED SOLID DIAGRAM
Toy (Cone mounted on a Hemisphere)
Total Surface Area of Toy = CSA of Cone + CSA of Hemisphere
TSA = πrl + 2πr² = πr(l + 2r)
TSA = πrl + 2πr² = πr(l + 2r)
03 · SOLVED NCERT EXAMPLE
Step-by-Step Problem Solving
EXAMPLE 1
A toy is in the form of a cone of radius 3.5 cm mounted on a hemisphere of same radius. The total height of toy is 15.5 cm. Find total surface area.
Radius r = 3.5 cm = 7/2 cm.
Height of hemisphere = radius r = 3.5 cm.
Height of cone h = Total Height - r = 15.5 - 3.5 = 12 cm.
Step 1: Slant Height l of Cone:
l = √(r² + h²) = √(3.5² + 12²) = √(12.25 + 144) = √156.25 = 12.5 cm
Step 2: Total Surface Area of Toy:
TSA = CSA of Cone + CSA of Hemisphere
TSA = πr(l + 2r)
TSA = (22/7) × (7/2) × [12.5 + 2(3.5)]
TSA = 11 × (12.5 + 7) = 11 × 19.5 = 214.5 cm²
TSA = πr(l + 2r)
TSA = (22/7) × (7/2) × [12.5 + 2(3.5)]
TSA = 11 × (12.5 + 7) = 11 × 19.5 = 214.5 cm²
04 · QUICK SUMMARY
Key Takeaways
01
Total Surface Area of combination = Sum of EXPOSED curved surface areas.
02
Slant height of cone: l = √(r² + h²).
03
Height of hemisphere is equal to its radius r.