Direct answer

What are the main sphere formulas?

The main sphere formulas are volume $V = \frac{4}{3} π r^{3}$ , surface area $S A = 4 π r^{2}$ , diameter $d = 2 r$ , and radius $r = d /2$ . If diameter is given, divide by 2 before using the volume or surface area formula.

Short answer

Sphere formulas at a glance

Volume: $V = \frac{4}{3} π r^{3}$ Surface area: $S A = 4 π r^{2}$ Diameter: $d = 2 r$

Definition

What is a sphere?

A sphere is a perfectly round 3D solid where every point on the surface is the same distance from the center.

Diagram

Sphere diagram

Sphere variables: radius r and diameter d.

Quick formula table

Sphere Formulas quick table

Formula	Equation	Variables
Volume	$V = \frac{4}{3} π r^{3}$	r = radius
Surface area	$S A = 4 π r^{2}$	r = radius
Diameter	$d = 2 r$	r = radius
Radius	$r = d /2$	d = diameter

Formula explanation

How to use these formulas

Use radius for both sphere volume and surface area. If you are given diameter, divide it by 2 first.

Calculator

Sphere Calculator

Enter measurements to get the result, unit, formula, and calculation steps.

Examples

Step-by-step examples

Find volume

Given r = 6 cm.
Use $V = \frac{4}{3} π r^{3}$ .
$V = 288 π \approx 904.78 c m^{3}$ .

Find surface area

Given r = 6 cm.
Use $S A = 4 π r^{2}$ .
$S A = 144 π \approx 452.39 c m^{2}$ .

Convert diameter to radius

Given d = 10 in.
Use $r = d /2$ .
$r = 10/2 = 5 in$ .

Find radius from volume

Given V = 36π in³.
Use $r = 3 3 V /4 π$ .
$r = 327 = 3 in$ .

Find radius from surface area

Given SA = 100π cm².
Use $r = S A /4 π$ .
$r = 25 = 5 c m$ .

Reverse calculations

Solve backwards from known values

Task	Formula	When to use it
Find radius from volume	$r = 3 3 V /4 π$	Use when volume is known.
Find radius from surface area	$r = S A /4 π$	Use when surface area is known.
Find diameter from volume	$d = 2 3 3 V /4 π$	Use when volume is known.

Common mistakes

Using diameter as radius in the volume formula.
Writing surface area in cubic units.
Forgetting that volume uses r cubed, while surface area uses r squared.

Practice

Practice problems with answers

1. Find volume when r = 3 cm.

$V = 36 π \approx 113.10 c m^{3}$ .

2. Find surface area when r = 3 cm.

$S A = 36 π \approx 113.10 c m^{2}$ .

3. Find radius when diameter is 14 m.

$r = 7 m$ .

4. Find radius when volume is 288π in³.

$r = 3 3 V /4 π = 3216 = 6 in$ .

5. Find diameter when surface area is 196π ft².

$r = 196 π /4 π = 7 f t$ , so $d = 14 f t$ .

Related formulas

FAQ

Sphere Formulas FAQ

What is the sphere volume formula?

The formula is $V = \frac{4}{3} π r^{3}$ .

What is the sphere surface area formula?

The formula is $S A = 4 π r^{2}$ .

How do you find radius from diameter?

Use $r = d /2$ .