Surface Area of a Sphere
Calculate the surface area of a sphere using SA = 4πr².
Find the outer surface area of balls, globes, and spherical objects.
The Formula
SA = 4 × π × r²
This formula calculates the total outer surface area of a sphere.
The surface area of a sphere is exactly 4 times the area of its great circle.
Variables
| Symbol | Meaning |
|---|---|
| SA | Surface area of the sphere |
| π | Pi, approximately 3.14159 |
| r | Radius of the sphere |
Example 1
Find the surface area of a sphere with radius 5 cm
SA = 4 × π × r² = 4 × π × 5²
SA = 4 × π × 25 = 100π
SA ≈ 314.16 cm²
Example 2
A globe has a diameter of 30 cm. How much material is needed to cover it?
Diameter = 30 cm, so radius = 15 cm
SA = 4 × π × 15² = 4 × π × 225 = 900π
SA ≈ 2,827.43 cm²
When to Use It
Use the surface area of a sphere formula when:
- Calculating how much material is needed to cover a ball or globe
- Determining paint or coating needed for spherical objects
- Working with heat transfer calculations (surface area affects cooling rate)
- Comparing the surface area of different sized spheres
Key Notes
- The sphere's surface area equals exactly 4 great circles (4πr²) — Archimedes proved this equals the lateral surface of its circumscribed cylinder, which he considered his greatest discovery
- From diameter: SA = π × d² — a convenient form that avoids the r = d/2 step
- Surface area scales as r² — doubling the radius quadruples the surface; for heat transfer and diffusion, smaller spheres have more surface area per unit of volume (surface-to-volume ratio = 3/r)
- Do not confuse with volume (4/3 πr³): surface area uses coefficient 4 and exponent 2; a useful memory aid is "4 circles" — the surface area is literally four times the area of a cross-section through the center