## Introduction

We have had a lot of rain in Minnesota this summer. As I sit here staring at the rain falling hard outside, I occurs to me that a cloud must be a very heavy thing in order to drop this much rain. It seems to me that figuring out the weight of a rain cloud would be a good Fermi problem. This cloud also dropped quite a bit of rain – how much of the cloud mass was lost as rain? Let’s dig in …

## Background

I need to round up some facts about clouds.

• Clouds float for the same reason a balloon floats – cloud material weighs less than air.
• The density of air at a typical altitude and temperature is about 1.007 kg/m3.
This is an interesting number. Air is heavier than I thought. Think about it – a cubic meter of air weighs 1.0 kg, which is 2.2 lbs. For some reason that seems like a lot to me.
• The density of cloud droplets is about 1.003 kg/m3.

Given these typical characteristics, lets try a specific example.

## A Cloud Example

The cloud that hangs over my home can be modeled as a cuboid.

• The cloud height (top to bottom) is 2 km (hCloud= 2 km)
• The cloud length is 10 km (lCloud= 10 km)
• The cloud width is 5 km (wCloud= 5 km)
• This cloud dropped 2 inches of rain (lRain= 2 inches)

## Analysis

A quick bit of Mathcad work gives me my estimate for cloud weight and the percentage of weight that rain represents.

Figure 1: Estimate of Cloud Weight and the Percentage Rain Represents of a Cloud.

## Conclusion

I would have never guessed that a cloud weighed that much. I also am surprised that a major rainfall represents such a small portion of the weight.

I am an engineer who encounters interesting math and science problems almost every day. I am not talking about BIG math here. These are everyday problems where a little bit of math really goes a long way. I thought I would write some of them down and see if others also found them interesting.
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### 5 Responses to How Much Does A Cloud Weigh?

1. CC says:

You calculated a much larger cloud than I did – I used a teeny little cotton-puff of a cloud to do my math on

• mathscinotes says:

I like your blog! I especially like your empirical approach to measuring the cloud. You sound like me. You encounter a lot of neat things in your reading and you want to share it. You now have a new blog follower.

mathscinotes

• CC says:

Well, I neither have nor know how to use MathCAD – it’s a skill I never had to develop. I was particularly proud of my astrolabe, however

I’m glad I found somebody else who likes to calculate random things just for the fun of it. Now to read more of your calculations!

• mathscinotes says:

Astrolabe? Cool! Could you post a photo? I just ordered a book with plans for some primitive navigation gear and I would like to build a project from the book.

As far as Mathcad goes, I am on a mission. I am a HUGE fan of computer algebra systems. I have found them very useful in my work and I use them for my recreational math as well. I use Mathcad as my main vehicle because it is the simplest to train people on.

2. CC says:

Ha, my “astrolabe” was pretty primitive, and I included a photo of it in the post. Basically a protractor with a weighted string clamped to it.

Because I don’t use MathCAD for work and I’m not a student… too bad they don’t have a “just playing with it” license with a price somebody who really is just playing with it could rationalize.