Question

jeremiah is using inscribed regular polygons to informally derive the formula for area of a circle. he draws a pentagon, octagon and decagon. as the number of sides, n, of the polygon increases, what does the value of a approach? as n→∞, a→□

Explanation:

Step1: Analyze the relationship between the polygon and the circle

As the number of sides \(n\) of the inscribed regular polygon increases, the polygon gets closer and closer to the circle. The apothem \(a\) of the regular polygon is the distance from the center of the polygon to the mid - point of one of its sides.

Step2: Consider the limit as \(n\to\infty\)

When \(n\to\infty\), the inscribed regular polygon approaches a circle. For a circle of radius \(r\), if we consider the geometric relationship between the apothem \(a\) of the inscribed regular polygon and the radius \(r\) of the circum - circle. As \(n\) becomes extremely large, the apothem \(a\) of the inscribed regular polygon approaches the radius \(r\) of the circle.

Answer:

\(r\)