# Unit shapes and a wealth of calculus problems

## TL;DR

This paper explores the concept of 'unit shapes' as analogs to the unit circle across various families of similar shapes, presenting calculus problems, isoperimetric questions, and new perspectives on mathematical constants.

## Contribution

It introduces the idea of 'unit shapes' as a natural and intrinsic concept within shape families and applies this to calculus and isoperimetric problems, offering new insights.

## Key findings

- Identification of 'unit shapes' as analogs to the unit circle
- Formulation of calculus problems related to extremal properties of unit shapes
- Revisiting isoperimetric problems using the concept of unit shapes

## Abstract

For a given family of similar shapes, what we call a "unit shape" strongly analogizes the role of the unit circle within the family of all circles. Within many such families of similar shapes, we present what we believe is naturally and intrinsically unital about their unit shapes. We present a number of calculus problems related to extremal questions about collections of unit shapes, and we recapitulate some isoperimetric problems in terms of unit shapes. We close by presenting some problems (some of which are open) and by proffering perhaps a new perspective on the $\pi$ vs. $\tau$ debate.