1. LESSON THIRTY: ‘ROUND ABOUT

2. INSCRIBED AND CIRCUMSCRIBED FIGURES We already know a bit about circles. You know how to find the radius. You know how to find the diameter. Using these, we found the circumference and area.

3. INSCRIBED AND CIRCUMSCRIBED FIGURES But if I gave you the area or circumference, could you find the radius or diameter? In essence, could you do what we’ve been doing…backwards?

4. INSCRIBED AND CIRCUMSCRIBED FIGURES

9. Now that we have learned this, we can learn a bit more vocabulary about circles. For instance, when two circles share a center, we call them concentric circles.

10. INSCRIBED AND CIRCUMSCRIBED FIGURES When a figure is perfectly inside a circle as in the figure below, we say that it is inscribed in the circle. This is to say that each of the figure’s vertices are on the circle.

11. INSCRIBED AND CIRCUMSCRIBED FIGURES We can also say that the circle is circumscribed around a polygon.

12. INSCRIBED AND CIRCUMSCRIBED FIGURES You will be required to find and draw an inscribed and circumscribed circle in various polygons. How will we do this? Any ideas.

13. INSCRIBED AND CIRCUMSCRIBED FIGURES Firstly, we must find the center. In a regular polygon, this is relatively easy. Simply enough, you draw two or more lines of symmetry and their point of intersection.

14. INSCRIBED AND CIRCUMSCRIBED FIGURES To draw a circle that is inscribed, our circle must touch all the sides of the polygon. How do we do this? Well, we simply take the length on of our apothem and make that the radius of our circle.

15. INSCRIBED AND CIRCUMSCRIBED FIGURES

16. When finding the circle that circumscribes the polygon, we use the radius of the polygon as our radius of the circle.

17. INSCRIBED AND CIRCUMSCRIBED FIGURES For triangles, we must recall some old knowledge. We must remember incenters and circumcenters.

18. INSCRIBED AND CIRCUMSCRIBED FIGURES You will recall that incenters are the intersections of all the angle bisectors of a triangle. One is pictured below. C

19. INSCRIBED AND CIRCUMSCRIBED FIGURES The circumcenter is the point of concurrence of the perpendicular bisectors of the sides of a triangle.

20. INSCRIBED AND CIRCUMSCRIBED FIGURES The circumcenter is equidistant from the vertices of the triangle. The incenter is equidistant from the sides of the triangle. You can use these to construct a circle around the triangle.

21. INSCRIBED AND CIRCUMSCRIBED FIGURES With the incenter as the circle’s center and the length to the sides as your radius, you can create a circle inscribed in your triangle.

22. INSCRIBED AND CIRCUMSCRIBED FIGURES With the circumcenter as the circle’s center and the length to the angles as your radius, you can create a circle circumscribed around your triangle.

23. INSCRIBED AND CIRCUMSCRIBED FIGURES The last thing I’ll ask you to know is how to construct a regular hexagon inscribed in a circle…given only a point and radius. The following video will show you step by step.

24. INSCRIBED AND CIRCUMSCRIBED FIGURES This one isn’t too difficult! http://www.mathopenref.com/constinhexag on.html http://www.mathopenref.com/constinhexag on.html

25. INSCRIBED AND CIRCUMSCRIBED FIGURES This is a lot of material so we’ll take our time with it! Do not panic!