11.1: Space Figures and Cross Sections
Polyhedron: solid that is bounded by polygons
Faces: polygons that enclose a polyhedron
Edge: line segment that faces meet and form
Vertex: point or corner where three or more edges meet
Example: The following is an example of a polyhedron. Fill the blanks with the appropriate
answer.
B
F
Faces:
A
Edges:
C
G
Vertices:
D
H
Example: Decide whether the solid is a polyhedron. If so, count the number of faces, vertices,
and edges. If not, explain why.
1)
Types of Solids (Not all are polyhedrons)
1) Prism
2) Pyramid
3) Cone
2)
4) Cylinder
5) Sphere
Regular Polyhedron: if all faces are congruent polygons
Convex polyhedron: if any two point on its surface can be connected by a segment that lies
entirely inside of on the polyhedron
Cross Section: a plane slicing through a solid, the intersection is a cross section
Example: Describe the shape formed by the cross section.
Platonic Solids: five regular polyhedra
tetrahedron (4 faces)
cube (6 faces)
octahedron (8 faces)
dodecahedron (12 faces)
icosahedron (20 faces)
Euler’s Theorem
The number of faces (F), vertices (V), and edges (E) of a polyhedron are related by the formula:
F+V=E+2
Example: Calculate the unknown given the information.
1) Faces:
2) Faces: 4
Edges: 12
Edges: 14
Vertices: 6
Vertices:
3) Faces: 7
Edges :
Vertices: 12
11.2: Surface Area of Prisms and Cylinders
Prism: polyhedron with two congruent faces, called bases
Lateral Faces: other faces of a prism that are not the two parallel congruent bases
Right Prism: each lateral edge is perpendicular to both bases
Oblique Prism: lateral edges not perpendicular to the bases
Slant Height: length of an oblique lateral edge
Diagrams
Right Prism
Oblique Prism (Draw one in)
Surface and Lateral Area of a Right Prism
S.A. = 2B + Ph (L + Ph) and L.A. = Ph
B = area of the base, P = perimeter of base, h = height of prism
Example: Find the surface area of the following.
P=
B=
h=
P=
B=
h=
Cylinder: solid with congruent circular bases
Right Cylinder: segment joining centers of each circle is perpendicular to bases
Diagram
Surface and Lateral Area of Cylinders
S = 2  r2 + 2  rh
and
L = 2  rh
r = radius of a base and h = height of a cylinder
Example: Find the surface area of the cylinders shown.
r=
h=
r=
h=
11.3: Surface Area of Pyramids and Cones
Pyramid: polyhedron in which the base if a polygon and the lateral faces are triangles
Height: distance from top of pyramid straight down to base
Slant Height: distance from top down the center of a face
Surface and Lateral Area of a Regular Pyramid
S.A. = B + ½ P ℓ (B + L.A.)
and
L. A. = ½ P ℓ
B = area of base, P = perimeter of base, ℓ = slant height of the pyramid
Example: Find the surface area of the pyramids shown.
B=
P=
ℓ=
B=
P=
ℓ=
B=
P=
ℓ=
Circular cone: cone that has a circular base and a vertex
Surface and Lateral Area of a Cone
S =  r2 +  r ℓ
and
Ex: Find the surface area of the following.
r=
ℓ=
r=
ℓ=
L=rℓ
11.4: Volume of Prisms and Cylinders
Volume of a Cube
V = s3
Volume of a Prism
V = Bh
Example: Find the volume of the following shapes.
Volume of a Cylinder
V =  r2 h
Example: Find the volume of the following cylinders.
11.5 Notes
Volumes of Pyramids and Cones
Volume of a Pyramid
1
V = Bh
3
Volume of Cones
1
 r2 h
V=
3
Remember that h = height, this is the distance from the top to the center of the base.
Example: Find the volume of the following.
11.6 Notes
Surface Area and Volume of Spheres
Sphere: locus of all point in space equidistant from a given point, the center
Surface Area of a Sphere
S = 4  r2
Volume of Spheres
V=
4
 r3
3
Example: Find the surface area and volume of the following spheres.
11.7: Areas and Volumes of Similar Solids
Similar Solids: solids with the same shape and all corresponding dimensions are proportional
Scale Factor: the ratio of two corresponding dimensions
Examples: Are the two figures similar? If so, give the scale factor of the first figure to the second figure.
1.
2.
Examples: Each pair of figures is similar. Use the given information to find the scale factor of the smaller
figure to the larger figure.
3.
4.
5.
Examples: The surface areas of two similar figures are given. The volume of the larger figure is given. Find the
volume of the smaller figure. (You must simplify before finding roots)
6. S.A. = 36 m2
7. S.A. = 108 in. 2
2
S.A. = 225 m
V = 750 m3
S.A. = 192 in.2
V = 1408 in.3
Examples: The volumes of two similar figures are given. The surface area of the smaller figure is given. Find
the surface area of the larger figure.
8. V = 8 m3
9. V = 125 in.3
3
V = 27 m
S.A. 36 = m2
V = 216 in.3
S.A. = 200 in.2