Lesson 12.1 Exploring Solids

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# Lesson 12.1 Exploring Solids

## Lesson 12.1 Exploring Solids

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1. Lesson 12.1 Exploring Solids Today, we will learn to… > use vocabulary associated with solids > use Euler’s Formula

2. Polyhedron ~ a solid formed by polygons

3. Prisms name the base are polyhedra named by their bases

4. Prisms have 2 parallel bases, Pyramids have 1 base.

5. Cylinders, Cones, & Spheres with curved surfaces are NOT polyhedra

6. Parts of Solids Faces Edges Vertex Bases vs Lateral Faces Height

7. + = 6 8 14 F + V = E = 12

8. + = 5 5 10 F + V = E = 8

9. Euler’s Theorem # Faces + # Vertices = # Edges + 2 F + V = E + 2

10. 1. In a polyhedron, F = 8 V = 10 E = ? find the number of edges. F + V = E + 2 8 + 10 = E + 2 18 = E + 2 ___ edges 16

11. 2. In a polyhedron, F = 5 V = ? E = 8 find the number of vertices. F + V = E + 2 5 + V = 8 + 2 5 + V = 10 ___ vertices 5

12. 3. In a polyhedron, F = ? V = 6 E = 9 find the number of faces. F + V = E + 2 F + 6 = 9 + 2 F + 6 = 11 5 ___ faces

13. A soccer ball is a polyhedron with 32 faces (20 hexagons & 12 pentagons). How many vertices does this polyhedron have? hexagons pentagons 20(6) 12(5) = 60 = 120 # Edges = ½ (120 + 60) = 90 32 + V = 90 + 2 V = 60 vertices

14. A polyhedron can be convex or concave.

15. 4. Describe the cross section shown. square

16. 5. Describe the cross section shown. pentagon

17. 6. Describe the cross section shown. triangle

18. A polyhedron is regular if all of its faces are congruent regular polygons. Platonic Solids

19. Lesson 11.1 Angle Measures in Polygons Today, we are going to… > find angle measures in polygons

20. 1 2 3 4 6 n-2 540˚ 720˚ (n-2)180˚ 1080˚ 180˚ 360˚ 60˚ 90˚ (n-2)180 n 108˚ 120˚ 135˚

21. Theorem 11.1 Polygon Interior Angles Theorem The sum of the measures of the interior angles is ____________ (n - 2)180°

22. (n - 2) 180° n Corollary to Theorem 11.1 The measure of one interior angle in a regular polygon is…

23. 1. Find the sum of the measures of the interior angles of a 30-gon. (30 - 2) 180 (28) 180 5040˚

24. 2.How many sides does a polygon have if the sum of the interior angles is 3240˚? 180(n-2) = 3240 n – 2 = 18 n = 20 sides

25. 1260˚ 9 (9 - 2)180 9 = (7)180 9 3. What is the measure of each interior angle of a regular nonagon? 140

26. 4. Find x. How many angles? 7 Sum? 180(7 - 2) 180(5) 900˚ 900° – given angles = x 900 – 783 = 117

27. 5. Find x. How many angles? 5 Sum? 180(5 - 2) 180(3) 540˚ sum of all angles = 540 43x – 19 = 540 x = 13

28. Exploring Exterior Angles GSP 1 2 5 3 4

29. Theorem 11.2 Polygon Exterior Angles Theorem The sum of the measures of one set of exterior angles in any polygon is _________ 360°

30. 360° n The measure of one exterior angle in any regular polygon is

31. 360° ext.  The number of sides in any regular polygon is… # sides =

32. One exterior angle and its interior angle are always ________________. supplementary

33. 360˚ 10 6. What is the measure of each exterior angle in a regular decagon? 36˚

34. 360˚ 40° 7. How many sides does a regular polygon have if each exterior angle measures 40˚? = 9 sides

35. 8. How many sides does a regular polygon have if each interior angle measures 165.6˚? 165.6 1 180 (n-2) n = 360˚ 14.4 Don’t write this down, yet. First, find the measure of an exterior angle. 180 – 165.6 = 14.4˚ 180(n – 2) = 165.6 n = 25 sides 180n – 360 = 165.6 n – 360 =– 14.4 n n =25 sides

36. 9. Find x. ? 70 90 ? 3x + 90 + 70 + 80 + 60 = 360° x = 20

37. 180(3) 5 180(4) 6 A soccer ball is made up of 20 hexagons and 12 pentagons. What’s the measure of each interior angle of a regular pentagon? = 108° What’s the measure of each interior angle of a regular hexagon? = 120°

38. Lesson 11.2 Today, we are going to… > find the perimeter and area of regular polygons Perimeter & Areas of Regular Polygons

39. (10) (5 3 ) 2 5 3 25 3 5 1. Find the area of this equilateral triangle 10 10 ≈ 43.3 60˚ 10 10

40. (24) (12 3 ) 2 12 3 144 3 2. Find the area of this equilateral triangle 24 24 ≈ 249.4 24

41. 2 s 2 3 Equilateral Triangles Area = s s s

42. apothem side side Area = ½ side · apothem · # sides side · # sides Perimeter =

43. number of isosceles triangles area of each isosceles triangle Area of Regular Polygons A =½(side)(apothem)(# sides) A = ½san

44. Perimeter of Regular Polygons P =(side)(# sides) P = sn

45. Find the area of the polygon. 3. a pentagon with an apothem of 0.8 cm and side length of 1.2 cm A = ½ (s) (a) (n) A = ½ (1.2) (0.8) (5) A = 2.4 cm2

46. (s)(n) Find the area of the polygon. 4. a polygon with perimeter 120 m and apothem 1.7 m A = ½ (s) (a) (n) A = ½ (1.7) (120) A = 102 m2

47. 5. Find the central angle of the polygon. 360° 5 360° 6 360° 8 360° 2(5) 360° 2(6) 360° 2(8) 6. Find the measure of this angle 36° 30° 22.5° 72° 60° 45°