limits

The triangle inequality has a familiar cadence to it and most students can recite it spontaneously. In this article, we mathematise our understanding of possible triangle shapes, using the limits of values which the angles first, and then the sides, take. It's a great way for students to explore different ways of expressing their conceptual understanding.

The approximation of π is a popular pastime of students of mathematics and I have started with the familiar age-old way, of fitting a regular polygon tightly in a circle of radius r. The vertices of the n-sided polygon are joined to the centre of the circle so that the angle subtended at the centre by each segment is 360/n. If n is sufficiently large, the perimeter of the polygon approaches the circumference 2πr of the circle and this approximation improves as the number of segments increases.

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