Geometry
Volume of a Sphere
Calculate the space inside any sphere.
The Formula
V = (4/3)πr³
Used for: Finding the volume (space inside) of a sphere
VVolume of the sphere
πPi ≈ 3.14159...
rRadius
🎯 Key Concepts
Understanding sphere volume:
🔮
r³ (cubed)Volume uses radius cubed, not squared
🔢
4/3 ≈ 1.33The coefficient is about 1.33 times π
📐
Units are cubedIf radius is in cm, volume is in cm³
📝 Worked Examples
Given Radius
Problem:
Find volume of sphere with r = 3 cm1
Formula
V = (4/3)πr³2
Substitute
V = (4/3) × π × 3³3
Calculate r³
V = (4/3) × π × 274
Simplify
V = 36π ≈ 113.1 cm³Answer: V ≈ 113.1 cm³
Given Diameter
Problem:
Diameter = 10 cm. Find volume.1
Find radius
r = d/2 = 5 cm2
Apply formula
V = (4/3)π × 5³3
Calculate
V = (4/3)π × 125 = 500π/34
Evaluate
V ≈ 523.6 cm³Answer: V ≈ 523.6 cm³
Finding Radius
Problem:
Volume = 288π. Find radius.1
Start with formula
(4/3)πr³ = 288π2
Solve for r³
r³ = 288 × (3/4) = 2163
Cube root
r = ∛216 = 6Answer: r = 6
💡 Pro Tips
Double r = 8× volume
Volume uses r³, so doubling radius multiplies volume by 8!
Quick estimate
V ≈ 4r³ is a rough mental estimate
Surface area relation
V = (4/3)πr³ while SA = 4πr² — notice the pattern!