Question

Group Activity Area of a Circle Inscribe a regular \(n-\) n-sided polygon inside a circle of radius 1 and compute the area of the polygon for the following values of \(n .\) (a) 4 (square) (b) 8 (octagon) (c) 16 (d) Compare the areas in parts (a), (b), and (c) with the area of the circle.

Short answer

The areas of the polygons inscribed in a circle of radius 1 are approximately 4 for a square, 5.53 for an octagon and 5.85 for a 16-sided polygon. The area of the circle itself is \(\pi\) or approximately 3.14. The areas of the polygons get closer to the area of the circle as the number of sides increases. This means that an \( n-sided \) polygon inscribed in a circle seems to 'fit' the circle more closely as \( n \) increases.

Step-by-step solution

Understand Formula of Regular Polygon Area

The formula for a regular polygon's area is: \(\frac{n \cdot s^2}{4 \cdot tan(\pi/n)}\) where n is the number of sides and s is the side length. Because the polygon is inscribed in a circle of radius 1, the side length s can be calculated using trigonometry.

Compute Polygon Area with n=4 (Square)

The square's side length is calculated as \(s = 2 \cdot sin(\pi/4) = \sqrt{2}\). Substituting into the area formula, the area is: \( \frac{4 \cdot (\sqrt{2})^2}{4 \cdot tan(\pi/4)} = 4\).

Compute Polygon Area with n=8 (Octagon)

The octagon's side length is calculated as \(s = 2 \cdot sin(\pi/8)\). Substituting into the area formula, we get the area of the octagon.

Compute Polygon Area with n=16

The 16-sided polygon's side length is calculated as \(s = 2 \cdot sin(\pi/16)\). Substituting into the area formula, the area is calculated.

Compare Areas

The area of the circle, given by \( \pi \cdot radius^2 = \pi \cdot (1)^2 = \pi \), is compared with the areas calculated in the previous steps.