Approximating Pi

ID: hudad-zuruz
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Created by Illustrative MathematicsIllustrative Mathematics, CC BY 4.0
Subject: Geometry
Grade: 9-12
Standards: HSG-GMD.A.1HSG-SRT.CHSN-Q.A.2
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11 questions

Approximating Pi

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More Sides

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

1) Area of the shaded region = ?

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2) Area of the shaded region = ?

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N Sides

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

3) Perimeter of the polygon = ?

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4) Perimeter of the polygon = ?

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5) Perimeter of the polygon = ?

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

So Many Sides

We can approximate the value of π\pi by inscribing an nn-sided polygon in a circle of radius 1 and calculating the perimeter of the polygon. The formula for the approximation is: πnsin(3602n)\pi \approx n \cdot \sin{\left(\frac{360}{2n}\right)}. Use this formula to find out how far from π\pi these approximations are for the given values of nn. Round your answers to three decimal places.

7) n=6n = 6

8) n=10n = 10

9) n=20n = 20

10) n=50n = 50

11) What value of nn approximates the value of π\pi to the thousandths place?