---
tag: 0WMDV48VSN
kind: statement
role: proposition
name: Volume of a ball in R^3
aliases: [volume-of-a-sphere]
msc: [26B15]
uses: [05WKRC7HWT, 00YH28R457, 0GTZ2F3PQY]
sources:
  - url: https://en.wikipedia.org/wiki/Ball_(mathematics)
    locator: Volume
  - url: https://en.wikipedia.org/wiki/Multiple_integral
    locator: Spherical coordinates
  - url: https://en.wikipedia.org/wiki/Cavalieri%27s_principle
    locator: History
---

## Statement

Let $R > 0$, and let $B = \{(x, y, z) \in \mathbb{R}^3 : x^2 + y^2 + z^2 \le R^2\}$.
Then
$$\operatorname{vol}(B) = \frac{4}{3} \pi R^3.$$

## Remarks

In [[spherical coordinates|05WKRC7HWT]], $\operatorname{vol}(B) = \int_0^{2\pi} \int_0^\pi \int_0^R \rho^2 \sin \varphi \, d\rho \, d\varphi \, d\theta = 2\pi \cdot 2 \cdot R^3 / 3$.
By slices, the cross-section at height $z$ is a disc of area $\pi (R^2 - z^2)$, and $\int_{-R}^{R} \pi (R^2 - z^2) \, dz = \tfrac{4}{3} \pi R^3$. This is [[Cavalieri's principle|0GTZ2F3PQY]].
This is the case $n = 3$ of the [[volume of a ball in R^n|00YH28R457]], since $\Gamma(5/2) = \tfrac{3}{4} \sqrt{\pi}$.
Archimedes found this volume, by comparison with a cone and a cylinder.

## Explanations

### L1 human

The ball of radius $R$ fills two thirds of the cylinder of radius $R$ and height $2R$ around it, since that cylinder has volume $2 \pi R^3$.
Any of the standard methods of multiple integration gives the same value.