Chapter 12 Notes
12.1 – Exploring Solids
A polyhedron is a solid bounded by polygons. Sides are faces,
edges are line segments connecting faces, and vertices are points
where the edges meet. The plural of polyhedron is polyhedra or
polyhedrons.
A polyhedron is regular if all the faces are congruent regular
polygons.
A polyhedron is convex if any two points on its surface can be
connected by a segment that lies entirely on the polyhedron (like
with polygons)
There are 5 regular polyhedra, called Platonic solids (they’re just
friends). They are a (look at page 721):
tetrahedron (4 triangular faces)
a cube (6 square faces)
octahedron (8 triangular faces)
dodecahedron (12 pentagonal faces)
icosahedron (20 triangular faces)
http://personal.maths.surrey.ac.uk/st/H.Bruin/image/PlatonicSolids.gif
Is it a polyhedron? If so, count the faces, edges, and vertices.
Also say whether or not it is convex.
No
Euler’s Theorem:
(Like FAVE two, or cube method)
Edges = Sides/2
Find vertices of polyhedra
made up of 8 trapezoids,
2 squares, 4 rectangles.
Find vertices of polyhedra
made up of 2 hexagons, 6
squares.
The intersection of a plane crossing a solid is called a cross
section. Sometimes you see it in bio, when they show you the
inside of a tree, the circle you get when you slice a tree is called
the cross section of a tree.
12.2 – Surface Area of Prisms
and Cylinders
Terms
P  Perimeter of one base
B  Area of base
b  base (side)
h  Height (relating to altitude)
l  Slant Height
TA  Total Area (Also SA for surface Area)
LA  Lateral Area
V  Volume
A prism has two parallel bases.
Altitude is segment perpendicular
to the parallel planes, also
referred to as “Height”.
Lateral faces are faces that are not
the bases. The parallel segments
joining them are lateral edges.
Prism
Net View
If the lateral faces are rectangles, it is called a
RIGHT PRISM. If they are not, they are called an
OBLIQUE PRISM.
Altitude.
RIGHT PRISM
OBLIQUE PRISM.
Lateral edge not
an altitude.
Lateral Area of a right prism is = bh + bh + bh + bh
the sum of area of all the
= (b + b + b + b)h
LATERAL faces.
LA = Ph
Total Area is the sum of ALL
the faces.
TA = 2B + Ph
Cylinder
LA  2 rh  Ph
SA  2 r  2 rh  2 B  LA
2
Find the LA, SA of this triangular prism.
LA =
SA =
8
4
3
5
Find the LA, SA of this rectangular prism.
LA =
5
SA =
3
8
Find the LA and TA of this regular hexagonal prism.
10
4
If it helps to think like
this.
Find the LA, and TA of this prism.
24 in
10 in
6 in
8 in
30 in
Find lateral area, surface area
Height = 8 cm
Height = 2 cm
Radius = 4 cm
Radius = .25 cm
Find the Unknown Variable.
SA = 192 units
2
x
2
8
V =
2x
x
4
72 units
3
Find the Unknown Variable.
SA = 40π cm2
SA = 100π cm2
Radius = 4 cm
Radius = r
Height = h
Height = 4 cm
12.3 – Surface Area of
Pyramids and Cones
vertex
Lateral
edge
Altitude
(height)
Slant
height
Lateral Face (yellow)
Base (light blue)
A regular pyramid has a regular polygon for a base
and its height meets the base at its center.
Lateral Area of a
regular pyramid is the
area of all the
LATERAL faces.
1
2
bl +
1
2
1
2
bl +
1
2
b l+
Total area is area of
bases. TA = B + 1 Pl
2
1
2
bl +
1
2
bl
(b + b + b + b + b) l =
1
2
Pl
Cone
LA   rl 
1
Pl
2
SA   r   rl  B  LA
2
Find Lateral Area, Total Area of regular hexagonal pyramid.
10 in
16 in
Find Lateral Area, Total Area of regular square pyramid.
13 cm
10 cm
Find lateral area, surface area
Find surface area.
Units in meters.
6
8
Slant Height = 15 in.
Radius = 9 in
Find unknown variable
8 cm
x cm
Slant height 8 cm
Slant height 8 in
Radius = ?
Radius = ?
TA = 105 cm2
TA = 48 π in2
Pyramid height 8 in
Slant height
2 cm
20 in
2 cm
12 in
Look at some cross sections
12.1 – 12.3 – More Practice,
Getting Ready for next week
Find the length of the unknown side
Find the area of the figures below, all shapes regular
Find the area of the shaded part
To save time, formulas for 12.4 are as follows:
Volume
of Prism
V  Bh
Volume
of cylinder
V  r h
2
12.4 – Volume of Prisms and
Cylinders
Altitude is segment
perpendicular to the
parallel planes, also
referred to as “Height”.
Volume of a right prism
equals the area of the
base times the height of
the prism.
V = Bh
Prism
The volume of an OBLIQUE PRISM is also Bh,
remember, it’s h, not lateral edge
Altitude.
RIGHT PRISM
OBLIQUE PRISM.
Lateral edge not
an altitude.
Find the V of this regular hexagonal prism.
10
4
Find the V of this triangular prism.
V
8
4
3
5
=  1 ( 3 )( 4 )  8  48 units
2

3
Find Volume
Height = 8 cm
Radius = 4 cm
Cylinder
V   r h  Bh
2
Circumference of a cylinder is 12π, and the height
is 10, find the volume.
Find the unknown variable.
What is the volume of the solid below?
What is the volume of the solid below? Prism below is a cube.
12.5 – Volume of Pyramids
and Cones
The volume
V 
1
of a pyramid
Bh
is :
Cone
V 
3
1
 r h
2
3
Slant Height = 15 in.
13 cm
Radius = 9 in
10 cm
1
3
Bh
Circumference of a cone is 12π, and the slant
height is 10, find the volume.
Find the height
an equilatera
of a pyramid
l triangle
with
base if the
side length is 8 in and the volume
96 3 in
3
is
Hexagon is regular, the box is
not. Hexagon radius 4 units,
height is 6 units Finding the
volume of box with hexagonal
hole drilled in it.
Find volume. Units
in meters.
8
12
Find Volume
Pyramid height 8 in
20 in
12 in
12.6 – Surface Area and
Volume of Spheres
A sphere with center O and
radius r is the set of all
points in SPACE with
distance r from point O.
Great Circle: A plane that
contains the center of a
circle.
Hemisphere: Half a sphere.
Chord: Segment whose
endpoints are on the sphere
Diameter: Segment through
center of the sphere
SA  4  r
V 
4
 r
2
3
3
Find SA and V with radius
6 m.
Radius of
Sphere
Circumference Surface Area
of great circle
of Sphere
Volume of
sphere
3m
4π in2
6π cm
9π ft3
2
Find the area of the cross section between the sphere and the
plane.
Radius 4 in. Cylinder
height 10 in. Find
area, volume
Find volume, side
length of cube is 3 in.
12.7 – Similar Solids
Find the total area and volume of a cube
with side lengths:
Area
1
2
5
10
Volume
Two shapes are similar if all the the sides have the same
scale factor.
If the scale factor of two similar solid is a:b, then
The ratio of the corresponding perimeters is a:b
The ratio of the base areas, lateral areas, and total
areas is a2:b2
The ratio of the volumes is a3:b3
Given the measure of the solids, state whether or not they are similar, and
if so, what the scale factor is.
Surface
area of A
Surface
Volume of Volume of Scale
Area of B A
B
Factor
100
144
125
216
4
9
64
125
1
4
8
27
Two similar cylinders have a scale factor of 2:3. If the
volume of the smaller cylinder is 16π units3 and the
surface area is 16π units2, then what is the surface
area and volume of the bigger cylinder?
Two similar hexagonal prisms have a scale factor of 3:4.
The larger hexagon has side length 4 in and height 9
in. Find the surface area and volume of the smaller
prism using ratios.