01 · RECAP FORMULAS
Volumes of Basic 3D Solids
Cylinder
Cylinder Volume
Volume = πr²h.Cone
Cone Volume
Volume = (1/3) πr²h (exactly 1/3rd of a cylinder of same radius and height!).Sphere / Hemisphere
Sphere & Hemisphere Volume
Sphere Volume = (4/3) πr³. Hemisphere Volume = (2/3) πr³.02 · COMBINED VOLUME RULE
Volume Additivity Principle
DIRECT SUM RULE FOR VOLUMES
Unlike surface areas (where touching internal faces are excluded), total volume is simply the sum of individual volumes!
Total Volume of Combined Solid = Volume(Solid 1) + Volume(Solid 2) + ...
COMBINED SOLID DIAGRAM
Gulab Jamun (Cylinder with two Hemispherical Ends)
Total Volume = Volume of Cylinder + 2 × (Volume of Hemisphere)
Total Volume = πr²h + 2 × [(2/3) πr³] = πr² [h + (4/3)r]
Total Volume = πr²h + 2 × [(2/3) πr³] = πr² [h + (4/3)r]
03 · SOLVED NCERT EXAMPLE
Step-by-Step Problem Solving
EXAMPLE 1
A gulab jamun contains sugar syrup up to about 30% of its volume. Find how much syrup would be found in 45 gulab jamuns, each shaped like a cylinder with two hemispherical ends with length 5 cm and diameter 2.8 cm.
Diameter = 2.8 cm ⟹ Radius r = 1.4 cm.
Height of cylinder h = Total Length - 2r = 5 - 2(1.4) = 5 - 2.8 = 2.2 cm.
Step 1: Volume of 1 Gulab Jamun:
V₁ = πr²h + (4/3)πr³ = πr² [h + (4/3)r]
V₁ = (22/7) × 1.4 × 1.4 × [2.2 + (4/3)(1.4)]
V₁ = 6.16 × [2.2 + 1.867] = 6.16 × 4.067 = 25.05 cm³
V₁ = (22/7) × 1.4 × 1.4 × [2.2 + (4/3)(1.4)]
V₁ = 6.16 × [2.2 + 1.867] = 6.16 × 4.067 = 25.05 cm³
Step 2: Total Volume of 45 Gulab Jamuns:
V_total = 45 × 25.05 = 1127.25 cm³
Step 3: Volume of Sugar Syrup (30%):
Syrup Volume = 30% × 1127.25 = (30 / 100) × 1127.25 = 338.18 cm³ ≈ 338 cm³
04 · QUICK SUMMARY
Key Takeaways
01
Volume of combined solid = Direct sum of individual volumes.
02
Cylinder height in capsule = Total Length - 2 × radius.
03
Cone volume is exactly (1/3)rd of a cylinder of same radius and height.