# Approximating Pi

11 questions

# Approximating Pi

##### More Sides

Calculate the area of the shaded regions. Leave answers in terms of $\pi$ and simplified radicals.

1) Area of the shaded region = ?

2) Area of the shaded region = ?

##### N Sides

Calculate the perimeter of each polygon. Leave answers in terms of a trigonometric function.

3) Perimeter of the polygon = ?

4) Perimeter of the polygon = ?

5) Perimeter of the polygon = ?

6) Come up with a general formula for the perimeter of the polygon in terms of $n$. Explain or show your reasoning.

##### So Many Sides

We can approximate the value of $\pi$ by inscribing an $n$-sided polygon in a circle of radius 1 and calculating the perimeter of the polygon. The formula for the approximation is: $\pi \approx n \cdot \sin{\left(\frac{360}{2n}\right)}$. Use this formula to find out how far from $\pi$ these approximations are for the given values of $n$. Round your answers to three decimal places.

7) $n = 6$

8) $n = 10$

9) $n = 20$

10) $n = 50$

11) What value of $n$ approximates the value of $\pi$ to the thousandths place?