Students explore how a regular polygon can be used to approximate the perimeter and area of a circle. The purpose of the activity is to provide a practical, geometric approach to understanding the concept of a limit.
Understand the concept of a limit. This activity is an ideal precursor for: ACMMM077.
Limit, infinite, inscribed, circumscribed,
About the Lesson
The notion of the ‘gradient at a point’ is difficult to comprehend for many students. Functions themselves are still relatively abstract concepts so differentiation from first principles and subsequent understanding of how this can be used to determine the gradient at a point is very challenging for students to understand. This activity uses a more tangible approach from the geometry world where using a regular polygon to estimate the circumference of a circle has clearly defined limits. Students produce equations to explore the limit, these equations are used to approximate the circumference of a circle in much the same way as differentiation is used to approximate the gradient at a point. The rules for the polygons are much more complicated that the rule for the circle where exact results are easily produced, in much the same way as common rules for differentiation are much simpler than using a limiting approach. The final question in this investigation can also be used to help understand one of the rules for differentiability with regards to approaching the same limit from either side.