1. Concept 52
3-D Shapes

2. Polyhedron
Solid
Convex
Concave
Regular
Cylinder
Prism
Cone
Pyramid
Sphere

3. Polyhedron: a solid figure with many plane faces

4. Solid: a 3-D shape that encloses space but is not made up of all polygon sides.

5. Convex: all vertices of the solid push outward.
Concave: one or more vertices of the solid are pushed inward.

6. Regular: a polyhedron with all the same regular polygons.

7. Prism: a polyhedron made up of two parallel bases connected by rectangles.
Rectangular prism
Triangular Prism
Pentagonal Prism

8. Pyramid:

9. Cylinder:

10. Cone:

11. Sphere:

12. Determine whether each solid is a polyhedron or solid
Determine whether each solid is a polyhedron or solid. Then draw a net for each if possible.
Pentagonal Prism
Cone
Sphere
Triangular Prism

13. Given the net of a solid. Draw the solid and give its name.5. 6. 7.
Triangular Prism
Hexagonal Pyramid
Cube

14. Parts of Solids and Cross Section
Concept 53
Parts of Solids and Cross Section

15. Parts of a 3D Shape! Fold on all dotted lines. (fold both directions)
Only cut this solid line

16. Edge: a segment where two faces come together.
Base: a polygon
Face: a set of polygons that make up the other surfaces of a polyhedron. (lateral faces)
Edge: a segment where two faces come together.
Vertex: a point where three or more edges come together.
Vertex
Edge
Edge
Edge
Face
Face
Vertex
Base:
Edge
Edge
Face
Edge
Vertex
Vertex
Triangular Pyramid

17. Then identify the solid
Then identify the solid. If it is a polyhedron, name the faces, edges, and vertices. 9.
10.
Faces:
Edges:
Vertices:
Pentagons: PWXYX and QRSUV,
Quadrilaterals: QVXW, UVXY, USZY, PRSZ, PRXW
Faces:
Edges:
Vertices:
Circle S (B)
none
Point R
𝑃𝑊 , 𝑊𝑋 , 𝑋𝑌 , 𝑌𝑍 , 𝑍𝑃 , 𝑃𝑅 , 𝑊𝑄 , 𝑋𝑉 , 𝑌𝑈 , 𝑍𝑆 , 𝑄𝑉 , 𝑉𝑈 , 𝑈𝑆 , 𝑆𝑅 , 𝑅𝑄
Points: P,W,X,Y, Z,Q,R,S,U,V

18. Then identify the solid
Then identify the solid. If it is a polyhedron, name the faces, edges, and vertices.
11.
Faces:
Edges:
Vertices:
Triangles: ABC and DEF,
Quadrilaterals: ABED, BCFE, ACFD
𝐴𝐵 , 𝐵𝐶 , 𝐶𝐴 , 𝐴𝐷 , 𝐵𝐸 , 𝐶𝐹 , 𝐷𝐸 , 𝐸𝐹 , 𝐹𝐷 ,
Points: A, B, C, D, E, F

19. Cross Section:
a surface or shape that is or would be exposed by making a straight cut through something, especially at right angles to an axis.
Sketch the cross section from a vertical slice of each figure. 1. 2. 3.

20. Describe each cross section.4. 5. 6.
Square
Triangle
Rectangle 7. 8.
Oval
Rectangle

22. Concept 54
Euler’s Theorem

23. There are 11 polyhedrons located around the room at each group of desks. Use each one to fill in a row of the table. If the shape has a name you know write it in the first column, otherwise just write what it is made up of. Ex. (2 triangles and 3 rectangles)

24. Name or what shapes make it.
# of Faces
# of Vertices
# of Edges 1 2 3 4 5 6 7 8 9 10 11 12

25. Euler’s Theorem
F + V = E + 2

26. Examples: 8 faces and 18 edges 21 edges and 14 vertices 8+𝑉=18+2
8+𝑉=20
𝑉=12
F+14=21+2
F+14=23
𝐹=9

27. 1 hexagon and 6 triangle faces
12 pentagon faces
8 triangle faces
1 hexagon and 6 triangle faces
12+𝑉=30+2
12∗5=𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑑𝑔𝑒𝑠
60=𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑑𝑔𝑒𝑠
12+𝑉=32
60 2 =𝑒𝑑𝑔𝑒𝑠 𝑤ℎ𝑒𝑛 𝑝𝑢𝑡 𝑡𝑜𝑔𝑒𝑡ℎ𝑒𝑟
30=
𝑉=20
8+𝑉=12+2
8∗3 2 =𝑒𝑑𝑔𝑒𝑠
8+𝑉=22
12=𝑒𝑑𝑔𝑒𝑠
𝑉=14
7+𝑉=12+2
1∗ ∗3 2 =𝑒𝑑𝑔𝑒𝑠
7+𝑉=22
3+9=12=𝑒𝑑𝑔𝑒𝑠
𝑉=15

28. 12 pentagon and 20 hexagon faces
20 triangle faces
12 pentagon and 20 hexagon faces
20+𝑉=30+2
20∗3 2 =𝑒𝑑𝑔𝑒𝑠
20+𝑉=32
30=𝑒𝑑𝑔𝑒𝑠
𝑉=12
12∗ ∗6 2 =𝑒𝑑𝑔𝑒𝑠
32+𝑉=90+2
32+𝑉=92
30+60=90=𝑒𝑑𝑔𝑒𝑠
𝑉=60

29. 2 hexagons and 6 rectangles
2∗ ∗6 2 =𝑒𝑑𝑔𝑒𝑠
6+12=18=𝑒𝑑𝑔𝑒𝑠
8+𝑉=18+2
8+𝑉=20
𝑉=12

30. Volume of Prisms
Concept 55

31. Rectangular Prism Triangular Prism Trapezoidal Prism Other Prisms
Volume = Base Area ∙ height
V = B ∙ h
Triangular
Prism
Trapezoidal
Prism
Other
Prisms

32. Rectangular Prisms V = B ∙ h V = B ∙ h V = (9 ∙5) ∙ 4 V = (6 ∙11) ∙ 2

33. Triangular Prism V = B ∙ h V = B ∙ h V = ( 1 2 10∙5) ∙ 14

34. Trapezoidal Prism V = B ∙ h V = B ∙ h V = ( 1 2 ∙2(4+7)) ∙ 7

35. Other Prisms V = B ∙ h V = B ∙ h V = ( 1 2 ∙(5.2)(6)(6)) ∙ 7

36. Find the volume of the right prism.
V = B ∙ h
V = B ∙ h
V = B ∙ h
V = (12∙6) ∙8
V = (2∙2) ∙8
V = ( 1 2 ∙4∙3) ∙3
V = 𝑚 3
V = 32 𝑐𝑚 3
V = 18 𝑖𝑛 3

37. Find the missing side length given the Volume, V of each solid.
4. V = 480 cm2
5. V = 120 in2
6. V = 180 cm2
V = B ∙ h
V = B ∙ h
V = B ∙ h
480 = (12∙𝑥) ∙8
120 = ( 1 2 ∙𝑥∙5) ∙4
180 = ( 1 2 ∙5(𝑥+12)) ∙4
480 = 96x
120 = 10x
180 =10(x +12)
12 in = x
18 = x + 12
5 cm = x
6 cm = x

38. Volume of Pyramids
Concept 56

39. Volume of Pyramids
Square
Pyramid
Triangular
Pyramid
Other
Pyramids

40. Volume of Pyramids
Volume = ∙ Base Area ∙ height
V = ∙ B ∙ h

41. Square Pyramids V = 1 3 ∙ B ∙ h V = 1 3 ∙ B ∙ h V = 1 3 ∙(7∙3)∙ 81. 2.
V = ∙ B ∙ h
V = ∙ B ∙ h
V = ∙(7∙3)∙ 8
V = ∙(10∙10)∙ 7
V = 56 𝑐𝑚 3
V = 𝑘𝑚 3

42. Triangular Pyramids V = 1 3 ∙ B ∙ h V = 1 3 ∙ B ∙ h3. 4.
V = ∙ B ∙ h
V = ∙ B ∙ h
V = ∙( 1 2 ∙4∙6)∙ 7
V = ∙( 1 2 ∙3∙4)∙5
V = 28 𝑐𝑚 3
V = 10 𝑐𝑚 3

43. Other Pyramids V = 1 3 ∙ B ∙ h V = 1 3 ∙ B ∙ h5. 6.
V = ∙ B ∙ h
V = ∙ B ∙ h
V = ∙( 1 2 ∙4.1∙5∙6)∙12
V = ∙(5∙4)∙6
V = 246 𝑓𝑡 3
V = 40 𝑐𝑚 3

44. Find the volume of each pyramid
Find the volume of each pyramid. Round to the nearest tenth if necessary.
𝑎 2 + 𝑏 2 = 𝑐 2
6 2 + 𝑥 2 = 10 2
V = ∙ B ∙ h
V = ∙ B ∙ h
36+ 𝑥 2 =100
V = ∙(3∙3)∙7
V = ∙(12∙8)∙10
𝑥 2 =64
𝑥=8
V = 21 𝑖𝑛 3
V = 320 𝑓𝑡 3
V = ∙ B ∙ h
V = ∙( 1 2 ∙6∙8)∙15
V = 120 𝑓𝑡 3

45. V = ∙ B ∙ h
V = ∙( 1 2 ∙6∙8)∙15
V = 120 𝑓𝑡 3

46. Volume of Cylinders, cones, and spheres
Concepts

52. Find the volume of each.

54. 7. hemisphere: area of
great circle ≈ 4π ft2