Key Concepts
- 1Volume of cone
- 2Volume of sphere
- 3Volume of hemisphere
- 4TSA of hemisphere
- 5CSA of hemisphere
- 6Frustum volume
- 7Slant height of cone
- 8CSA of cone
- 9How do you find the surface area of a combination of two solids?
- 10What is the formula for the volume of a frustum of a cone?
Important Formulas & Facts
#1
V = (1/3)πr²h
#2
V = (4/3)πr³
#3
V = (2/3)πr³
#4
3πr² (curved + base)
#5
2πr²
#6
V = (πh/3)(r₁² + r₂² + r₁r₂)
#7
l = √(h² + r²)
#8
πrl
#9
Add the surface areas of the individual solids, then subtract the areas of the surfaces that are joined together (hidden/overlapping surfaces).
#10
V = (1/3) x pi x h x (r1^2 + r2^2 + r1*r2), where r1 and r2 are the radii of the two circular ends and h is the height.
Must-Know Questions
Q1500 persons take a dip in a cuboidal pond 80 m long and 50 m broad. Average water displacement per person is 0.04 m³. Find the rise in water level.
Explanation
Total displacement = 500 × 0.04 = 20 m³. Rise = Volume/(length × breadth) = 20/(80 × 50) = 20/4000 = 0.005 m = 0.5 cm.
Q2Cone on hemisphere, both radius 1 cm, cone height 3 cm. Find volume.
Explanation
(1/3)π(1)²(3) + (2/3)π(1)³ = π + 2π/3 = 5π/3 cm³.
Q3Sphere of radius 4.2 cm melted into cylinder of radius 6 cm. Find height.
Explanation
(4/3)π(4.2)³ = π(36)h. h = (4/3)(74.088)/36 = 2.744 cm.
Q4TSA of hemisphere of radius 7 cm:
Explanation
TSA = 3πr² = 3(22/7)(49) = 462 cm².
Q5Cylinder to cone volume ratio (same base and height):
Explanation
πr²h : (1/3)πr²h = 3:1.
Practice Surface Areas and Volumes
Reinforce what you just revised with practice questions