Sphere Formulas (Volume, Surface Area)

A sphere is the set of all points a fixed distance r from a center in three dimensions. Its volume is V = 4⁄3·πr³ and its surface area is SA = 4πr² — exactly four great circles.

Sphere formulas

Volume: $V = \tfrac{4}{3}\pi r^3$
Surface area: $SA = 4\pi r^2$

where r = radius, V = volume, SA = surface area

Volume

V = \tfrac{4}{3}\pi r^3

Worked example

A sphere of radius 3 has volume 36π ≈ 113.10 cubic units.

V = \tfrac43\pi r^3 = \tfrac43\pi\cdot 27 = 36\pi \approx 113.10

Surface area

SA = 4\pi r^2

Worked example

Its surface area is also 36π — a coincidence unique to radius 3.

SA = 4\pi r^2 = 4\pi\cdot 9 = 36\pi \approx 113.10

Interactive sphere explorer

Drag a slider (or type a value, or use the arrow keys) and watch the diagram, the results, and the full step-by-step substitution update live.

Radius r

Volume V 113.1units³

Surface area SA 113.1units²

Step-by-step substitution

V = 4⁄3 πr³ = 4⁄3 · π · 3³ = 36π ≈ 113.1
SA = 4πr² = 4 · π · 3² = 36π ≈ 113.1

Classroom use: sphere formulas

Who it's for: Grades 8–12 · Grade 8 and high-school students learning sphere volume and surface area.

Learning objectives

After using this page, students will be able to:

Compute sphere volume using V = 4⁄3πr³.
Compute surface area using SA = 4πr².
Explain why volume grows with the cube of the radius and area with the square.

Suggested classroom activity

At r = 3 both volume and surface area equal 36π — a coincidence worth exploiting. Ask students to find it in the explorer and explain why it happens at that radius and nowhere else; it forces genuine attention to the exponents.

Standards alignment

Aligned to the Common Core State Standards for Mathematics. Codes are given exactly as published so they can be matched against state crosswalks.
Code	Standard
`8.G.C.9`	Know and use the formulas for the volumes of cones, cylinders, and spheres.
`HSG.GMD.A.3`	Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Free to use and adapt in class under the terms on our license page — no account, no sign-up, no ads. Content last reviewed August 2026.

Related & next steps

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Frequently asked questions

What is the volume formula for a sphere?

A sphere’s volume is V = 4⁄3·πr³. A sphere of radius 3 has volume 4⁄3·π·27 = 36π ≈ 113.10 cubic units.

What is the surface area of a sphere?

A sphere’s surface area is SA = 4πr², four times the area of a great circle. A sphere of radius 5 has surface area 4π·25 = 100π ≈ 314.16.

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