Volume of Cylinders, Cones, and Spheres

Volume of a Cylinder

Cylinder: 3D shape with circular base and top, straight sides

Formula: V = πr²h

Where:

• r = radius of circular base
• h = height
• π ≈ 3.14159

Think: Area of base (πr²) × height

Example 1: Basic Cylinder

Radius = 3 cm, Height = 10 cm

Calculate:

V = πr²h
V = π(3)²(10)
V = π(9)(10)
V = 90π cm³
V ≈ 282.7 cm³

Answer: 90π cm³ or ≈ 282.7 cm³

Example 2: Given Diameter

Diameter = 8 m, Height = 5 m

Find radius first:

r = d/2 = 8/2 = 4 m

Calculate volume:

V = π(4)²(5)
V = π(16)(5)
V = 80π m³
V ≈ 251.3 m³

Answer: 80π m³

Example 3: Solve for Height

Volume = 150π ft³, Radius = 5 ft, Find h

Use formula:

150π = π(5)²h
150π = 25πh
h = 150π/(25π)
h = 6 ft

Answer: h = 6 ft

Volume of a Cone

Cone: 3D shape with circular base, comes to a point

Formula: V = (1/3)πr²h

Where:

• r = radius of base
• h = height (perpendicular from base to apex)

Remember: Cone volume is 1/3 of cylinder with same base and height

Example 1: Basic Cone

Radius = 6 cm, Height = 9 cm

Calculate:

V = (1/3)πr²h
V = (1/3)π(6)²(9)
V = (1/3)π(36)(9)
V = (1/3)(324π)
V = 108π cm³
V ≈ 339.3 cm³

Answer: 108π cm³

Example 2: Cone vs Cylinder

Compare volumes: same radius 4 m, same height 12 m

Cylinder:

V = π(4)²(12) = 192π m³

Cone:

V = (1/3)π(4)²(12) = (1/3)(192π) = 64π m³

Cone is 1/3 of cylinder!

Example 3: Given Diameter

Diameter = 10 in, Height = 6 in

Radius = 5 in

Calculate:

V = (1/3)π(5)²(6)
V = (1/3)π(25)(6)
V = 50π in³
V ≈ 157.1 in³

Answer: 50π in³

Volume of a Sphere

Sphere: Perfectly round 3D shape (like a ball)

Formula: V = (4/3)πr³

Where:

• r = radius

All points on sphere are same distance from center

Example 1: Basic Sphere

Radius = 3 cm

Calculate:

V = (4/3)πr³
V = (4/3)π(3)³
V = (4/3)π(27)
V = 36π cm³
V ≈ 113.1 cm³

Answer: 36π cm³

Example 2: Given Diameter

Diameter = 12 ft

Radius = 6 ft

Calculate:

V = (4/3)π(6)³
V = (4/3)π(216)
V = 288π ft³
V ≈ 904.8 ft³

Answer: 288π ft³

Example 3: Solve for Radius

Volume = 288π m³, Find r

Use formula:

288π = (4/3)πr³
288 = (4/3)r³
864 = 4r³
r³ = 216
r = 6 m

Answer: r = 6 m

Comparing the Three Formulas

Cylinder: V = πr²h

Cone: V = (1/3)πr²h

Sphere: V = (4/3)πr³

Pattern: All involve π and powers of r

Hemisphere (Half-Sphere)

Hemisphere: Half of a sphere

Formula: V = (1/2) × (4/3)πr³ = (2/3)πr³

Example: Hemisphere Volume

Radius = 9 cm

Calculate:

V = (2/3)π(9)³
V = (2/3)π(729)
V = 486π cm³
V ≈ 1526.8 cm³

Answer: 486π cm³

Composite Shapes

Combine multiple shapes

Example 1: Cylinder with Cone on Top

Cylinder: r = 5 m, h = 10 m Cone: r = 5 m, h = 4 m

Cylinder volume:

V₁ = π(5)²(10) = 250π m³

Cone volume:

V₂ = (1/3)π(5)²(4) = (100π/3) m³

Total:

V = 250π + (100π/3)
V = (750π/3) + (100π/3)
V = (850π/3) m³
V ≈ 890.1 m³

Example 2: Sphere Inside Cylinder

Find volume of space between cylinder and sphere (both radius 6 cm, cylinder height 12 cm)

Cylinder:

V = π(6)²(12) = 432π cm³

Sphere:

V = (4/3)π(6)³ = 288π cm³

Space between:

V = 432π - 288π = 144π cm³

Real-World Applications

Cylindrical tanks: Water storage, fuel tanks

Cones: Ice cream cones, traffic cones, funnels

Spheres: Balls, planets, bubbles, globes

Hemispheres: Domes, bowls

Example 1: Water Tank

Cylindrical water tank: diameter 4 m, height 6 m

How many liters? (1 m³ = 1000 L)

Radius = 2 m

Volume:

V = π(2)²(6)
V = 24π m³
V ≈ 75.4 m³

Convert to liters:

75.4 × 1000 = 75,400 L

Answer: About 75,400 liters

Example 2: Basketball

Basketball diameter ≈ 24 cm. Find volume.

Radius = 12 cm

Volume:

V = (4/3)π(12)³
V = (4/3)π(1728)
V = 2304π cm³
V ≈ 7238.2 cm³

Example 3: Ice Cream Cone

Cone: radius 3 cm, height 10 cm Hemisphere scoop on top: radius 3 cm

Cone volume:

V₁ = (1/3)π(3)²(10) = 30π cm³

Hemisphere volume:

V₂ = (2/3)π(3)³ = 18π cm³

Total ice cream:

V = 30π + 18π = 48π cm³ ≈ 150.8 cm³

Units of Volume

Always cubed units: cm³, m³, in³, ft³

Conversions:

• 1 m³ = 1,000,000 cm³
• 1 ft³ = 1728 in³
• 1 m³ = 1000 liters

Practice