# Derivative of Area Equals Perimeter - Coincidence or Rule?

Rina Zazkis, Roza Leikin, Ilya Sinitsky

Research output: Contribution to journalArticlepeer-review

## Abstract

Why is the derivative of the area of a circle equal to its circumference? Why is the derivative of the volume of a sphere equal to its surface area? And why does a similar relationship not hold for a square or a cube? Or does it? In their work in teacher education, these authors have heard at times undesirable responses to these questions: "That's the way it is. Circles and spheres are very special. Squares and cubes have corners." Or, "It is a simple coincidence with circles. This relationship does not hold for any other shapes." This article explores and explains the familiar relationship of the area of a circle and its circumference and of the volume of a sphere and its surface area. It then extends this relationship to other two- and three-dimensional figures--squares and regular polygons, cubes and regular polyhedra.
Original language English 686-692 7 The Mathematics Teacher 106 9 Published - 1 May 2013

## Keywords

• Equations (Mathematics)
• Geometric Concepts
• Mathematical Concepts
• Mathematics Instruction
• Teacher Education

## Fingerprint

Dive into the research topics of 'Derivative of Area Equals Perimeter - Coincidence or Rule?'. Together they form a unique fingerprint.