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 sensemaking (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 Materials
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.
Learning Targets
Student Facing
 I can explain how to use regular polygons to approximate the value of $\pi$.
CCSS Standards
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.