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# Archimedes Solves Density Problem

By Ron Kurtus (updated 8 February 2022)

You need to know both the mass and volume of an object to determine its density. It is usually easy to find the mass of an object, but finding the volume of an irregularly shaped object can be a challenge.

This relates to is a famous story where ancient Greek scientist Archimedes was asked by King Hiero of Syracuse to find out if the gold wreath made by Hiero's goldsmith was truly pure gold and not mixed with some other alloy. The king suspected his goldsmith was embezzling some of the gold.

If the wreath was pure gold, it would have a certain density. If it was made of a mixture, the density would be different. However, in order to find the density of the wreath, its volume must be determined. This was the problem Archimedes faced.

After thinking about it for a while, Archimedes then took a bath to find a solution to the problem.

Questions you may have include:

• What is one way to find the volume?
• How did taking a bath help?
• What was the solution to the problem?

This lesson will answer those questions. Useful tool: Units Conversion

## One way to find the volume

One solution Archimedes proposed was to crush the irregularly shaped wreath into a cube to determine its volume. He could measure its mass to determine its density.

ρ = m/V

where

• ρ is the density of the wreath material (ρ is the Greek letter rho)
• m is the mass of the wreath
• V is its volume

The king would not accept that solution.

## Takes a bath

The story goes that Archimedes decided to take a hot bath to help his mind relax and find a solution to this problem. When he noticed the water rise as he got into the tub, Archimedes suddenly realized the solution.

Archimedes was so excited that he jumped out of the tub and ran down the street, shouting, "Eureka! Eureka!" which means "I have found it!" Unfortunately, he was so excited that he forget to put on his clothes and ran through the streets naked!

## Solution

The solution to the problem was that he placed the wreath in a container of water and measured its displacement. That could be done by filling the container to the rim, placing the wreath in the water, and then measuring the overflow.

By measuring the volume of the water and the mass of the wreath, Archimedes was able to determine its mass. Unfortunately for the goldsmitth, the density of the wreath showed that it was not pure gold. He was robbing the king and was probably punished for the crime.

## Summary

By realizing that an object submerged in water will displace its volume—no mater what it's shape or configuration—Archimedes was able to determine the volume of the wreath. Then by weighing it, he could determine its mass and thus its density.

By comparing the wreath's density with the known density of gold, Archimedes determined that the wreath was not made of pure gold.

Find different ways to solve your problems

## Resources and references

Ron Kurtus' Credentials

### Websites

Archimedes - Short biography

Density - Wikipedia

Archimedes - Wikipedia

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