1. Polygons and Measurement
Math 8 Unit 8
Polygons and Measurement
Strand 4: Concept 1 Geometric Properties
PO 2. Draw three-dimensional figures by applying properties of each
PO 3. Recognize the three-dimensional figure represented by a net
PO 4. Represent the surface area of rectangular prisms and cylinders as the area of their net.
PO 5. Draw regular polygons with appropriate labels
Strand 4: Concept 4 Measurement
PO 1. Solve problems for the area of a trapezoid.
PO 2. Solve problems involving the volume of rectangular prisms and cylinders.
PO 3. Calculate the surface area of rectangular prisms or cylinders.
PO 4. Identify rectangular prisms and cylinders having the same volume.

2. Key Terms
Def: Polygon: a closed plane figure formed by 3 or more segments that do not cross each other.
Def: Regular Polygon: a polygon with all sides and angles that are equal.
Def: Interior angle: an angle inside a polygon
Def: Exterior angle: an angle outside a polygon

3. Triangle 3 Sides 3 Angles Sum of Interior Angles 180
Each angle measures 60 if regular.

4. Quadrilateral 4 Sides 4 Angles Sum of Interior Angles 360
Each angle measures 90 if regular.

5. Pentagon 5 Sides 5 Angles Sum of Interior Angles 540
Each angle measures 108 if regular.

6. Hexagon 6 Sides 6 Angles Sum of Interior Angles 720
Each angle measures 120 if regular.

7. Heptagon 7 Sides 7 Angles Sum of Interior Angles 900
Each angle measures 128.6 if regular.

8. Octagon 8 Sides 8 Angles Sum of Interior Angles 1080
Each angle measures 135 if regular.

9. Nonagon 9 Sides 9 Angles Sum of Interior Angles 1260
Each angle measures 140 if regular.

10. Decagon 10 Sides 10 Angles Sum of Interior Angles 1440
Each angle measures 144 if regular.

11. Sum of Interior Angles =
Formula:
Sum of Interior Angles =
180 (n-2)

12. Example: Find the sum of the interior angles in the given polygon.
a. 14-gon
b. 20-gon
180(n-2)
180(n-2)
180(14-2)
180(20-2)
180●12
180●18
Total = 2160 
Total = 3240 

13. Example: Find the measure of each angle in the given regular polygon.
a. 16-gon
b. 12-gon
180(n-2)
180(n-2)
180(16-2)
180(12-2)
180●14
180●10
Total = 2520 
Total = 1800 
1800 ÷12
2520 ÷16
150 
157.5 

14. Formula:
Sum of Exterior Angles
Is always 360

15. Example: Find the length of each side for the given regular polygon and the perimeter.
a.) rectangle, perimeter 24 cm
24 ÷ 4
6 cm
b.) pentagon, 55 m
55 ÷ 5
11 m

16. Example: Find the length of each side for the given regular polygon and the perimeter.
c.) nonagon, 8.1 ft
8.1 ÷ 9
0.9 ft
d. heptagon, 56 mm
56 ÷ 7
8 mm

17. Perimeter
Evil mathematicians have created formulas to save you time. But, they always change the letters of the formulas to scare you!
Any shape’s “perimeter” is the outside of the shape…like a fence around a yard.

18. Perimeter
To calculate the perimeter of any shape, just add up “each” line segment of the “fence”.
Triangles have 3 sides…add up each sides length. 8 8
8+8+8=24
The Perimeter is 24 8

19. Perimeter
A square has 4 sides of a fence 12 12 12
=48 12

20. Regular Polygons Just add up EACH segment10
8 sides, each side 10 so =80

21. Area Area is the ENTIRE INSIDE of a shape
It is always measured in “squares” (sq. inch, sq ft)

22. Different Names/Same idea
Length x Width = Area
Side x Side = Area
Base x Height = Area

23. Notes 3-D Shapes
Base:
Top and/or bottom of a figure. Bases can be parallel.
Edge:
The segments where the faces meet.
Face:
The sides of a three-dimensional shape.
Nets: Are used to show what a 3-D shape would look like if we unfolded it.

24. Prisms
Have Rectangles for faces
Named after the shape of their Bases

25. More Nets

26. Prisms Fun with by D. Fisher
This activity can be done with powerpoint only, but it is much better if the students practice with paper prisms or models made of drinking straws, etc. Besides learning the names of various prisms students can practice “algebraic reasoning.” They can generate sequences and make predictions about patterns. Enjoy this presentation. Send feedback and suggestions to Don Fisher:
by D. Fisher

27. Vertices (points) 6 Edges (lines) 9 Faces (planes) 5
Triangular Prism
Vertices (points) 6
Edges (lines) 9
Faces (planes) 5
The base has 3 sides.

28. The base has sides. 4 Vertices (points) 8 Edges (lines) 12
Rectangular Prism
The base has sides. 4
Vertices (points) 8
Edges (lines) 12
Faces (planes) 6

29. The base has sides. 5 Vertices (points) 10 Edges (lines) 15
Pentagonal Prism
The base has sides. 5
Vertices (points) 10
Edges (lines) 15
Faces (planes) 7

30. The base has sides. 6 Vertices (points) 12 Edges (lines) 18
Hexagonal Prism
The base has sides. 6
Vertices (points) 12
Edges (lines) 18
Faces (planes) 8

31. The base has sides. 8 Vertices (points) 16 Edges (lines) 24
Octagonal Prism
The base has sides. 8
Vertices (points) 16
Edges (lines) 24
Faces (planes) 10

32. Pyramids Have Triangles for faces
Named after the shape of their bases.

33. Fun with Pyramids By D. Fisher
This activity can be done with powerpoint only, but it is much better if the students practice with paper pyramids or models made of drinking straws, etc. Besides learning the names of various pyramids students can practice “algebraic reasoning.” They can generate sequences and make predictions about patterns. Enjoy this presentation. Send feedback and suggestions to Don Fisher:
By D. Fisher

34. Vertices (points) 4 Edges (lines) 6 Faces (planes) 4
Triangular Pyramid
Vertices (points) 4
Edges (lines) 6
Faces (planes) 4
The base has 3 sides.

35. Vertices (points) The base has sides. 4 5 Edges (lines) 8
Square Pyramid
Vertices (points)
The base has sides. 4 5
Edges (lines) 8
Faces (planes) 5

36. Vertices (points) The base has sides. 5 6 Edges (lines) 10
Pentagonal Pyramid
Vertices (points)
The base has sides. 5 6
Edges (lines) 10
Faces (planes) 6

37. Vertices (points) The base has sides. 6 7 Edges (lines) 12
Hexagonal Pyramid
Vertices (points)
The base has sides. 6 7
Edges (lines) 12
Faces (planes) 7

38. Vertices (points) The base has sides. 8 9 Edges (lines) 16
Octagonal Pyramid
Vertices (points)
The base has sides. 8 9
Edges (lines) 16
Faces (planes) 9

39. Name Picture Base Vertices Edges Faces
Triangular Pyramid
Square Pyramid
Pentagonal Pyramid
Hexagonal Pyramid
Heptagonal Pyramid
Octagonal Pyramid 3 4 6 4 4 5 8 5 5 6 10 6 6 7 12 7
Draw it 7 8 14 8 8 9 16 9
No picture n
n + 1 2n
n + 1
Any Pyramid

40. Name Picture Base Vertices Edges Faces
Triangular Prism
Rectangular Prism
Pentagonal Prism
Hexagonal Prism
Heptagonal Prism
Octagonal Prism 3 6 9 5 4 8 12 6 5 10 15 7 6 12 18 8
Draw it 7 14 21 9 8 16 24 10
No picture n 2n 3n
n + 2
Any Prism

41. Cylinder
Circles for bases
Rectangle for side

42. Points of View View point is looking down on the top of the object.
up on the bottom of
the object.
View point is looking
from the right (or left)
of the object.

43. Example 1 :
Top
Front
Top View
Side
Front View
Side View
Front
Side
Bottom View
Bottom

44. Example 2 : Top D q H
Front View
Left View

45. Example 3 : Top view
Front View
Left View

46. Example 4
Top View
Front View
Left View
Bottom View

47. Surface Area
Surface Area: the total area of a three-dimensional figures outer surfaces.
Surface Area is measured in square units (ex: cm2)

48. Rectangular Prism
SA=2lw +2lh + 2wh l h w

49. 1. Find the surface area. SA=2lw +2lh + 2wh SA=248 + 242 + 282
SA= 112 cm2

50. SA=2lw +2lh + 2wh SA=266 + 2610 + 2610 SA= 72 + 120 + 120
2. Find the surface area of a box with a length of 6 in, a width of 6 inches and a height of 10 inches.
SA=2lw +2lh + 2wh
SA=266 + 2610 + 2610
SA=
SA= 312 cm2

51. Cylinder SA =2r2 + 2rh SA = r2 + r2 + hC r C=2r A=hC So A=2rh h

52. Examples: 1. Find the surface area.
SA =2r2 + 2rh
SA = 2(5)2 + 2(5)(20)
SA = 225 + 2100
SA = 50 
SA = 250 = 785 cm2

53. SA =2r2 + 2rh SA = 2(9)2 + 2(5)(18) SA = 281 + 290
2. Find the surface area of a cylinder with a height of 5 in and a diameter of 18 in.
SA =2r2 + 2rh
SA = 2(9)2 + 2(5)(18)
SA = 281 + 290
SA = 162 
SA = 342 = in2

54. Volume Volume: The amount of space inside a 3D shape.
Volume is measure in cubic units (ex: cm3)

55. Rectangular Prism
V=LWH
V = 842
V = 64 cm3

56. Cylinder V = 2(5)2(20) V = 22520 V = 2500 SA = 1000 = 3140 cm3
V=r2h
V = 2(5)2(20)
V = 22520
V = 2500
SA = 1000 = 3140 cm3

57. Triangular Prism
V= ½ LWH
V = ½  244910
V = 5, 880 cm3

58. Surface Area or Volume Covering a Triangular speaker box with carpet?
Filling a triangular speaker box with foam?
Volume
Filling a triangular box with M n M’s?

59. Surface Area or Volume Painting the outside of a triangular prism?
Covering a triangular piece of chocolate with paper?
Filling a triangular mold with concrete?
Volume