Exploring Polygons and Circles: Properties and Calculations

1. Chapter 8 Polygons and Circles Students will use properties of triangles to help determine interior and exterior angles of polygons(review). Students will discovery methods for finding areas of polygons. Students will discovery properties of circles.

2. Section 8.1 Students will discover special intersections points in relationship to triangles.

3. Special segments in Triangles Medians connects vertex to midpoint of side opposite Height or Altitude segment from vertex, perpendicular to side opposite Angle Bisectors Segment that bisects an angle and touches side opposite Perpendicular Bisector Segment that is the perpendicular bisector of a side Points of Concurrency Where the specific segments above intersect and what is special about them

4. Angle Bisector Incenter is the point of concurrency – where all three angles bisectors of a triangle intersect This point is equidistant from all the sides of the triangle You can use this point as the center of the circle and a point on the each side of the triangle to construct an inscribe circle Used to construct a circle inside the triangle – inscribed circle Always formed inside the triangle

5. Perpendicular Bisector Circumcenter – this is the point of concurrency, where all 3 perpendicular bisectors intersect This point is equidistant from all of the vertices If you use the circumcenter as the center of the circle and all the vertices as points on the circle you will construction a circumscribed circle about the triangle

6. Altitudes Orthocenter – where the three altitudes intersect Nothing sepecial about this point

7. Medians Centriod – is the point of concurrency for the medians of a triangle This is the point of the center of gravity for the triangle, could balance the triangle at this point The centroid of a triangle divides each median into 2 parts so that the distance from the centroid to the vertex is twice the distance from the centroid to the midpoint of the opposite side. Always formed inside the triagnle

8. Examples

9. Homework

10. Section 8.2 Students will review interior and exterior angles or polygons and discovery how to find areas of polygons.

11. Angles of Polygons How do we determine the sum of the interior angles of a polygon 180(n-2) What is the sum of the exterior angles of a polygon 360 How do we find each interior angle of a regular polygon Sum/n How do we find the measure of each exterior angle of a regular polygon 360/n

12. Area Amount of 2 dimensional space a figure takes up Triangle height is also the altitude (b*h)/2 Rectangle/Parallelograms B*h or L*W Square S^2

13. Trapezoid Think of this as two triangles – they have different bases Sum of two bases times height divided by 2 Can’t always divide trapezoid into a rectangle and two triangles, the end triangles may not be the same

14. Kite Split into 2 congruent triangles using the diagonals Area of one triangle times 2

15. Examples

16. Other polygons How would you find the area of a pentagon Non Regular or Regular Other Shapes

17. Find the area of this regular pentagon, you know the Perimeter is 200 inches.

19. Area • Divide the polygon into Triangles – one vertex is the center of the polygon and the side of the polygon is the base of the triangle – this form congruent isosceles triangles • The base = perim/n • Find the central angle of each isosceles triangle – angle formed at the center • 360/n • Draw the altitude of the isosceles triangle to make a right triangle, this bisect the central angle and the base • Calculate the altitude of the triangle using trig ratios Alt= ½ side/ tan (1/2 central angle) • Find the area of the triangle • Find the area of the polygon by multiplying by how many triangles there are

20. Apothem Apothem is the perpendicular distance from the center of the polygon to one side (height of the triangle) If you know the apothem you can find the area as well a=apothem s=side lengths n=number of sides

21. Examples

22. Examples

23. Similar Polygons and Areas Remember proportions and scale factor How do we use this to find areas of similar shapes

24. Homework worksheets

25. Section 8.4 Students will discovery properties of circles and how to calculate area and circumference.

26. Circles Exact answer or in terms of pi – do not multiply out the pi value Approximate value is when you use, 3.14, 22/7 or the pi button on calculator

27. Examples

28. Parts of Circles - Sector Chord – segment inside a circle touching two points Sector of a Circle – region between two radii and an arc of the circle, slice of pizza

29. Parts of Circle - Segment Segment of a circle – is the region between a chord and an arc of a circle, crust of pizza

30. Practice

31. Concentric Circles – same center different radius Annulus – the region between two concentric circles

32. Practice

33. More Practice

34. Homework worksheet