Your textbook has many, many problems on finding the measurements of a circle, so I looked for problems that are off the beaten track. The result is an unusual set of applications to the circle, therefore challenging but intriguing, I think, for middle school classes. Let your colleagues know of your own ideas  and comments on this topic. Just add a note below.

Discovering the Value of Pi
Students measure the diameter and circumference of several circles, using a handy applet, record their data and reach conclusions about the ratio of circumference to diameter. A genuine guided exploration!

Windshield Wipers: It’s Raining! Who Sees More? The Driver of the Car or the Truck?
In this activity, students compare the areas cleaned by different wiper designs. An animation shows the movement of the two windshield wipers, each cleaning off a different geometric shape on the window. Students are encouraged to draw the shape cleaned by each wiper and find its area.

The Great Circle
By clicking on two cities on a world globe, students see two line segments connecting the cities, one showing the great circle route (the shortest) and the other showing the route on a flat map. An interesting  and visual application of real-world math.

Three Piece Circle Puzzle
Students create the puzzle themselves, using compasses, and are challenged to find the area of each of the three pieces. You will need to guide your eighth- and ninth-grade students through the given solution.

The Noon Day Project: Measuring the Circumference of the Earth
Here is a real-world project that will engage your class in measuring the circumference of the Earth! You will find all information you need to enable students to recreate the measurement as done by the Greek librarian Eratosthenes over 2000 years ago. The procedure is based on measurements of shadows taken at high noon local time on a designated day; results from several schools are posted online and used to calculate the circumference. Included are detailed explanations and illustrations of the mathematics involved, including a video of Carl Sagan explaining Eratosthenes’ calculation.