1. Surface Area and Volume
Chapter 12

2. 12.1 Exploring Solids
Polyhedron: a solid that is bounded by polygons, called faces, that encolose a single region of space.
Edge: the line segment formed by the intersection of two faces.
Vertex: a point where 3 or more edges meet.
Regular: all faces are congruent regular polygons

3. Types of Solids Prism polyhedron Pyramid polyhedron
Cone not a polyhedron

4. Types of Solids
Cylinder not a polyhedron
Sphere not a polyhedron

5. Convex vs. Concave
Convex
Concave
(nonconvex)

6. Cross Sections
If a plane slices through a solid, it forms a cross section.
The cross section of the sphere and the plane is a circle.

7. Cross Sections
What shape is formed by the intersection of the plane and the solid?

8. Cross Sections
What shape is formed by the intersection of the plane and the solid?

9. Platonic Solids Regular Tetrahedron Cube Regular Octahedron
4 faces, 4 vertices, 6 edges
Cube
6 faces, 8 vertices, 12 edges
Regular Octahedron
8 faces, 6 vertices, 12 edges
Regular Dodecahedron
12 faces, 20 vertices, 30 edges
Regular Icosahedron
20 faces, 12 vertices, 30 edges

10. Euler’s Theorem
Faces + Vertices = Edges + 2

11. 12.2 & 12.4 Surface Area and Volume of Prisms and Cylinders
Prism: polyhedron with 2 congruent faces, called bases, that lie in parallel planes.
Lateral faces: parallelograms formed by connecting the bases.
Lateral edges: the segments connecting the vertices of the bases.

12. More Vocabulary
Right prism: each lateral edge is perpendicular to both bases
Oblique prisms: all prisms that are not right prisms

13. Surface Area
Find the surface area of a right rectangular prism with a height of 8in., a length of 3in., and a width of 5in.

14. Surface Area
The surface area (S) of a right prism can be found using the formula S = 2B + Ph, where B is the area of a base, P is the perimeter of a base, and h is the height.

15. Examples
Find the surface area of the right prism. 6 5 10

16. Examples
Find the surface area of the right prism. 7 7 5 7

17. Cylinders
A cylinder is a solid with congruent circular bases that lie in parallel planes.
Lateral area of a cylinder is the area of its curved surface.
Surface area is the sum of the lateral area and the areas of the two bases.

18. Surface Area of Cylinders
The surface area S of a right cylinder is S = 2B + Ch = 2pr2 + 2prh, where B is the area of the base, C is the circumference of the base, r is the radius of the base, and h is the height.

19. Examples
Find the surface area of the right cylinder.

20. Examples
Find the height of a cylinder with a radius of 6.5 and a surface area of

21. Volume of a Solid Volume of a Cube: V = s3
Volume Congruence Postulate:
If 2 polyhedra are congruent then their volumes are the same.
Volume Addition Postulate:
The volume of a solid is the sum of the volumes of all its non-overlapping parts.

22. Volume of a Prism
The volume (V) of a prism is V = Bh, where B is the area of a base and h is the height.

23. Examples
Find the volume of the prism.

24. Volume of a Cylinder
The volume V of a cylinder is V = Bh = pr2h, where B is the area of a base, h is the height, and r is the radius of a base.

25. Examples
Find the volume of the right cylinder.

26. Examples
Find the volume of the solid.

27. Examples
Find the volume of the solid.

28. 12.3 & 12.5 Surface Area & Volume of Pyramids & Cones
Pyramid is a polyhedron with a polygon base and triangular lateral faces with a common vertex.
A regular pyramid has a regular polygon for a base and its height meets the base at its center.

29. Surface Area of a Regular Pyramid
The surface area S of a regular pyramid is S = B + ½ PL, where B is the area of the base, P is the perimeter of the base, and L is the slant height.

30. Examples
Find the surface area of the regular pyramid.

31. Surface Area of a Cone
The surface area S of a right cone is S = pr2 + prL, where r is the radius of the base and L is the slant height.

32. Examples
Find the surface area of the cone.

33. Volume of a Pyramid
The volume V of a pyramid is V = 1/3Bh, where B is the area of the base and h is the height.

34. Example
Find the volume of a pyramid.

35. Volume of a Cone
The volume V of a cone is V = 1/3 Bh = 1/3pr2h, where B is the area of the base, h is the height, and r is the radius of the base.

36. Example
Find the volume of the Cone.

37. Examples
Find the volume of the solid.

38. Examples
Find the volume of the solid.

39. 12.6 Volume and Surface area of Spheres
Surface Area of a Sphere: S = 4pr2
Surface Area = S ; radius = r

40. Surface Area
Plane intersects a sphere and the intersection contains the center of the sphere, the intersection is a great circle.
Great circles divide the sphere into two hemispheres.
The equator is a great circle.

41. Using a Great Circle
C = 13.8p ft. for the great circle of a sphere. What is the surface area of the sphere?

42. 12.6
Volume of a Sphere: V = 4/3 pr3
V = volume ; r = radius

43. Examples Radius of Sphere Circum. Of great circle SA of sphere
Volume of Sphere
7mm ?
144p in2
10p cm
4000p m3 3

44. 12.7 Similar Solids
Two solids with equal ratios of corresponding linear measures are called similar solids.

45. Similar Solids Theorem
If 2 similar solids have a scale factor of a:b, then corresponding areas have a ratio of a2:b2, and corresponding volumes have a ratio of a3:b3.

46. Examples
The prisms are similar with a scale factor of 1:3. Find the surface area and volume of G if the surface are of F is 24 ft2 and the volume is 7 ft3. G F

47. Examples
Write the ratio of the two volumes.
V = 512 m3
V = 1728 m3