# Approximating Pi Illustrative Mathematics, CC BY 4.0
Subject: Geometry
Standards: HSG-GMD.A.1HSG-SRT.CHSN-Q.A.2

11 questions

# Approximating Pi

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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.

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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?