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GBK Geometry

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GBK Geometry

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  1. GBK Geometry Jordan Johnson

  2. Today’s plan • Greeting • Review Asg #85: • From Ch. 13, Lesson 1 (pp. 530-535): • Exercises 1-14, 46-48. • From Ch. 13, Lesson 2 (pp. 537-539): • Easy: 1-2, 6-12. Med: 29-31. • Bonus: Set III from Ch. 13, Lesson 1. • Lesson: Regular Polygons – Conclusion • Homework / Questions • Clean-up

  3. Regular Polygon Perimeter & Area

  4. Regular Polygon Perimeter & Area

  5. Regular Polygon Perimeter & Area

  6. Regular Polygon Perimeter & Area or 6r (or we could write r⁄2 instead) or 3⁄2 or (33⁄2)r2

  7. Regular Polygon Perimeter & Area

  8. How about a10,000-gon?

  9. A 10,000-gon • n = 10,000 ☺ • Central radius/apothem angle = 180°⁄10,000 • Side length = 2r sin(180°⁄10,000) • Apothem = r cos(180°⁄10,000) • Perimeter = 20,000r sin(180°⁄10,000) • Area = 10,000r2 sin(180°⁄10,000)cos(180°⁄10,000)

  10. How about an n-gon?

  11. 10,000-gon n-gon • Central radius/apothem angle = 180°⁄10000 = 180°⁄n • Side length = 2r sin(180°⁄10000) = 2r sin(180°⁄n) • Apothem = r cos(180°⁄10000) = r cos(180°⁄n) • Perimeter = 20,000r sin(180°⁄10000) = 2nr sin(180°⁄n) • Area = 10000r2 sin(180°⁄10000)cos(180°⁄10000) = nr2 sin(180°⁄n)cos(180°⁄n)

  12. Perimeter = 2nr sin(180°⁄n) Area = nr2 sin(180°⁄n)cos(180°⁄n) What happens to all thisas n gets reallybig?

  13. Perimeter = 2nr sin(180°⁄n) Area = nr2 sin(180°⁄n)cos(180°⁄n) What happens to all thisas n gets reallybig? Let’s isolate the parts with n…

  14. Perimeter = 2nr sin(180°⁄n) = 2(n sin(180°⁄n))r Area = nr2 sin(180°⁄n) cos(180°⁄n) = (n sin(180°⁄n) cos(180°⁄n))r2

  15. Perimeter = 2nr sin(180°⁄n) = 2(n sin(180°⁄n))r Area = nr2 sin(180°⁄n) cos(180°⁄n) = (n sin(180°⁄n) cos(180°⁄n))r2

  16. Calculator time…

  17. Calculator time…

  18. Calculator time…

  19. As N gets large… • n sin(180°⁄n) approaches… • n sin(180°⁄n) cos(180°⁄n) approaches… 

  20. Perimeter = 2(n sin(180°⁄n))r  2r Area = (n sin(180°⁄n) cos(180°⁄n))r2  r2

  21. Full Circle For a given radius r… The area of a circle is the limit thatthe area of an n-sided polygonapproaches as n grows. The circumference is the limit thatthe perimeter of an n-sided polygon approaches as n grows.

  22. Homework • Final review – due the day of the final. • Make a note card for the final. • Journal Entry #16: • What is one helpful thing you learned in geometry class about math this year? • What is one helpful thing you learned in geometry class that wasn't about math this year? • If you could change one thing about the class, what would it be? • If you could have done one thing differently, what would it be? • What advice would you give to another student who is going to take this class? • Due the end of finals week.

  23. Clean-up / Reminders • Pick up all trash / items. • Push in chairs (at front and back tables). • See you tomorrow!