Lesson 11

Approximating Pi

Lesson Narrative

In this lesson students build off their concrete calculations from the previous lesson to write a generalized formula for the perimeter of a polygon inscribed in a circle of radius 1. The relatively unstructured presentation of this activity is purposeful so students can build their perseverance and sense-making (MP1). Students should work with their groups to determine what information they need, how they calculated this information in the specific cases, and how they can express those repeated procedures in a generalized formula (MP8).

Once students build a generalized formula, they apply the formula to approximate the value of $$\pi$$. During the lesson synthesis students learn methods ancient mathematicians used to generate both upper and lower bounds of $$\pi$$. This lesson presents an opportunity for seeing how mathematics has changed over history and how ancient techniques are still in use but with the added power of computers.

Learning Goals

Teacher Facing

• Compare and contrast approximations of $\pi$ (orally).
• Explain how to use regular polygons to approximate the value of $\pi$ (using words and other representations).

Student Facing

• Let’s approximate the value of pi.

Required Preparation

Devices are required for the digital version of the activity N Sides. Acquire devices that can run the GeoGebra applet, ideally 1 per student.

Be prepared to display an applet for all to see during the lesson synthesis.

Student Facing

• I can explain how to use regular polygons to approximate the value of $\pi$.

Building Towards

Glossary Entries

• arccosine

The arccosine of a number between 0 and 1 is the acute angle whose cosine is that number.

• arcsine

The arcsine of a number between 0 and 1 is the acute angle whose sine is that number.

• arctangent

The arctangent of a positive number is the acute angle whose tangent is that number.

• cosine

The cosine of an acute angle in a right triangle is the ratio (quotient) of the length of the adjacent leg to the length of the hypotenuse. In the diagram, $$\cos(x)=\frac{b}{c}$$.

• sine

The sine of an acute angle in a right triangle is the ratio (quotient) of the length of the opposite leg to the length of the hypotenuse. In the diagram, $$\sin(x) = \frac{a}{c}.$$

• tangent

The tangent of an acute angle in a right triangle is the ratio (quotient) of the length of the opposite leg to the length of the adjacent leg. In the diagram, $$\tan(x) = \frac{a}{b}.$$

• trigonometric ratio

Sine, cosine, and tangent are called trigonometric ratios.