---
tag: 04SD3F4FCR
kind: problem
name: Doubling the cube
aliases: [delian-problem, duplication-of-the-cube]
msc: [51M15]
uses: [0DAECMZVRT]
sources:
  - url: https://en.wikipedia.org/wiki/Doubling_the_cube
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Doubling_the_cube
    locator: History
same_as:
  wikidata: Q213673
  wikipedia: Doubling the cube
---

## Problem

Given the edge of a cube, construct with [[straightedge and compass|0DAECMZVRT]] alone the edge of a cube with twice the volume.

## Remarks

For an edge of length $1$, the new edge has length $\sqrt[3]{2}$.
So the problem asks whether $\sqrt[3]{2}$ is a [[constructible number|0FMN2MN7TM]].
The answer is no: [[doubling the cube is impossible|0CQ357M0QS]].
The problem has solutions with other tools, such as a marked ruler or conic sections.

## History

The problem is also called the Delian problem, after a legend about an oracle on the island of Delos.
Hippocrates of Chios showed that it is equivalent to finding two mean proportionals between a segment and a segment twice as long.
Pierre Wantzel proved in 1837 that no straightedge and compass solution exists.

## Explanations

### L1 human

The problem asks to build a cube with twice the volume of a given cube, using only an unmarked ruler and a compass.
It is one of the three classical construction problems of Greek geometry.