Length of a Roll of Paper

Introduction

My father once asked me a similar question so he could understand how long a roll was likely to last based on the remaining diameter. How much paper is left on this roll? How could you estimate it?

Guiding Questions

• A roll of paper is a spiral of many turns. How can you estimate its total length from the inner and outer radius?
• Model the roll as many circles, then as an arithmetic series. What total length does the series give?
• Unroll and measure a real roll (till receipt, toilet roll). How close was your prediction?
• Can you do it with integration or an Archimedean spiral, and do the methods agree?