Geometry: An Interactive Journey to Mastery

Lesson 22: Explorations with Pi

Explorations with Pi

Lesson 22

Topics

x Pi.

x Circumference, arc length, and area formulas.

'H¿QLWLRQV

x DUFOHQJWK: The length of an arc of a circle.

x pi,QÀDWJHRPHWU\WKHUDWLRRIWKHFLUFXPIHUHQFHRIDFLUFOHWRLWVGLDPHWHULVWKHVDPHIRUDOOFLUFOHV

7KHFRPPRQYDOXHRIWKLVUDWLRLVFDOOHGSLDQGLVGHQRWHGʌ

x VHFWRURIDFLUFOH7KH¿JXUHIRUPHGE\WZRUDGLLRIDFLUFOHDQGDQDUFRIWKHFLUFOHEHWZHHQWKHP

)RUPXODV

x If C is the circumference of a circle, D is its diameter, rLVLWVUDGLXVͼ௘WKXVD = 2r௘ͽ

and A is its area, then

2

.

2.

.

CD

Cr

Ar

S

S

S

x If an arc of the circle of radius r has measure xƒWKHQWKHOHQJWKRIWKHDUFLV 360 x ˜2.Sr

x 7KHDUHDRIWKHVHFWRUGH¿QHGE\WKDWDUFLV 360 x ˜Sr^2.

Summary

We like to believe that the ratio of the circumference of a circle to its diameter is the same value for all circles.

In this lesson, we examine the basis of that belief and examine the arc length and area formulas that follow

from it.