Surface Area and Volume of a Sphere

Learn formulas and calculations for spheres. Math 10-C Alberta mathematics.

Lesson 1.5 of Measurement in Math 10C — Alberta curriculum lessons.

What you'll learn in this lesson

• Where the Formulas Come From. Discover how the sphere surface area formula SA = 4πr² follows from a cylinder wrapper, and how the volume formula V = (4/3)πr³ is exactly two-thirds of that same cylinder. Both derivations are built from shapes you already know.
• Sphere Examples. Apply SA = 4πr² and V = (4/3)πr³ in three worked examples: finding surface area from a circumference, computing the enormous volume of the sun, and working backwards from surface area to find a diameter range.
• Hemispheres. Learn why slicing a sphere creates an extra circular face, giving SA = 3πr² for a hemisphere. Apply both hemisphere formulas, then practise with four problems ranging from the moon of Endor to a basketball.

Inside the lesson — a free preview

The Big Idea

Sphere — Key Definition. A sphere is the set of all points in space that are the same distance from a centre point — like a perfectly round ball.

That fixed distance is called the radius r.

The distance all the way across the sphere through the centre is the diameter d = 2r.

Strategy: Both Formulas Use the Radius. Both sphere formulas require the radius, not the diameter.

If a problem gives you the diameter, your first step is always:

r = (d)/(2)

Surface Area: SA = 4πr²

Imagine wrapping a cylinder snugly around a sphere — same radius, and tall enough that the sphere just fits inside. The cylinder's height equals the sphere's diameter, which is 2r.

Derivation — Where SA = 4πr² Comes From. It turns out the sphere's surface area equals the curved (lateral) area of that surrounding cylinder.

The lateral area of the cylinder is circumference times height:

Step 1: Lateral area = 2π r × 2r

Step 2: Simplify:

A = 4π r^2

Volume: V = (4/3)πr³

A sphere's volume is exactly (2)/(3) of the cylinder that surrounds it (radius r, height 2r).

Derivation — Where V = (4/3)πr³ Comes From. Step 1: Volume of the cylinder = π r^2 × 2r = 2π r^3

Step 2: The sphere holds (2)/(3) of that cylinder:

V = (2)/(3) × 2π r^3

Step 3: Simplify:

V = (4)/(3)π r^3

Memory Tip — r² vs. r³. Surface area uses r^2 (area is two-dimensional).

Volume uses r^3 (volume is three-dimensional).

If a question gives the diameter, halve it to get r before doing anything else.

Worked example: Surface Area from Circumference

A satellite is wrapped in polyester film. How much film is needed to cover the Echo Satellite, which has a circumference of 95.8 m?

Strategy — Find Radius First. The problem gives the circumference, not the radius.

Use C = 2π r to solve for r first, then apply SA = 4π r^2.

Solution. Step 1: Use C = 2π r to find the radius:

95.8 = 2π r

r = (95.8)/(2π)

r = (95.8)/(6.283)

r ≈ 15.2 m

Step 2: Now use SA = 4π r^2:

SA = 4π(15.2)^2

SA = 4π(231.04)

Answer: SA ≈ 2903.3 m^2 of film is needed.

Key Tip. When a sphere problem gives the circumference instead of the radius, C = 2π r is your way in.

Find r first, then go to the SA or volume formula.

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