Assignment
• P. 806-9: 2-20
even, 21, 24, 25,
28, 30
• P. 814-7: 2, 3-21
odd, 22-25, 30
• Challenge
Problems: 3-5, 8, 9
In Glorious 3-D!
Most of the figures you have worked with so
far have been confined to a plane—twodimensional. Solid figures in the “real
world” have 3 dimensions: length, width,
and height.
Polyhedron
A solid formed by polygons that enclose a
single region of space is called a
polyhedron.
Separate your Geosolids into 2 groups: Polyhedra and others.
Parts of Polyhedrons
• Polygonal region = face
• Intersection of 2 faces = edge
• Intersection of 3+ edges = vertex
face
edge
vertex
Warm-Up
Separate your
Geosolid
polyhedra into
two groups
where each of
the groups
have similar
characteristics.
What are the names of
these groups?
Polyhedra: 1. Prisms
2. Pyramids
12.2 & 12.3: Surface Area of Prisms,
Cylinders, Pyramids, and Cones
Objectives:
1. To find and use formulas for the lateral
and total surface area of prisms,
cylinders, pyramids, and cones
Prism
A polyhedron is a prism
iff it has two congruent
parallel bases and its
lateral faces are
parallelograms.
Classification of Prisms
Prisms are classified by their bases.
Right & Oblique Prisms
Prisms can be right or oblique. What
differentiates the two?
Right & Oblique Prisms
In a right prism, the lateral edges are
perpendicular to the base.
Pyramid
A polyhedron is a pyramid iff it has one base
and its lateral faces are triangles with a
common vertex.
Classification of Pyramids
Pyramids are also classified by their bases.
Pyramid
A regular pyramid is one whose base is a
regular polygon.
Pyramid
A regular pyramid is one whose base is a
regular polygon.
• The slant height is the
height of one of the
congruent lateral faces.
Solids of Revolution
The three-dimensional figure formed by
spinning a two dimensional figure around
an axis is called a solid of revolution.
Cylinder
A cylinder is a 3-D
figure with two
congruent and
parallel circular
bases.
• Radius = radius of
base
Cone
A cone is a 3-D figure
with one circular base
and a vertex not on the
same plane as the
base.
• Altitude = perpendicular
segment connecting
vertex to the plane
containing the base
(length = height)
Cone
A cone is a 3-D figure
with one circular base
and a vertex not on the
same plane as the
base.
• Slant height = segment
connecting vertex to the
circular edge of the base
Right vs. Oblique
What is the difference between a right and
an oblique cone?
Right vs. Oblique
In a right cone, the segment connecting the vertex to
the center of the base is perpendicular to the base.
Nets
Imagine cutting a 3-D
solid along its
edges and laying
flat all of its
surfaces. This 2-D
figure is a net for
that 3-D solid.
An unfolded pizza box is a net!
Nets
Imagine cutting a 3-D
solid along its
edges and laying
flat all of its
surfaces. This 2-D
figure is a net for
that 3-D solid.
Activity: Red, Rubbery Nets
Match one of the red, rubbery nets with its
corresponding 3-D solid. Which of the
shapes has no net?
Activity: Red, Rubbery Nets
A sphere
doesn’t have
a true net; it
can only be
approximated.
Match one of the red, rubbery nets with its
corresponding 3-D solid. Which of the
shapes has no net?
Exercise 1
There are generally two types of
measurements associated with 3-D solids:
surface area and volume. Which of these
can be easily found using a shape’s net?
Surface Area
The surface area of a 3-D
figure is the sum of the
areas of all the faces or
surfaces that enclose the
solid.
• Asking how much
surface area a figure has
is like asking how much
wrapping paper it takes
to cover it.
Lateral Surface Area
The lateral surface area of
a 3-D figure is the sum
of the areas of all the
lateral faces of the solid.
• Think of the lateral
surface area as the size
of a label that you could
put on the figure.
Exercise 2
What solid corresponds to the net below?
How could you find
the lateral and total
surface area?
Exercise 3
Draw a net for the rectangular prism below.
A
B
To find the lateral surface area,
you could:
• Add up the areas of the lateral rectangles
C
D
Exercise 3
Height of Prism
Draw a net for the rectangular prism below.
Perimeter of the Base
To find the lateral surface area,
you could:
• Find the area of the lateral surface as one, big rectangle
Exercise 3
Height of Prism
Draw a net for the rectangular prism below.
Perimeter of the Base
To find the total surface area,
you could:
• Find the lateral surface area then add the two bases
Surface Area of a Prism
Lateral Surface Area
of a Prism:
S  Ph
• P = perimeter of the base
• h = height of the prism
Total Surface Area of a
Prism:
S  Ph  2B
• B = area of the base
Exercise 4
Find the lateral and total surface area.
Exercise 5
Draw a net for the cylinder.
Notice that the lateral surface of a cylinder is also a
rectangle. Its height is the height of the cylinder,
and the base is the circumference of the base.
Exercise 6
Write formulas for the lateral and total surface area
of a cylinder.
Surface Area of a Cylinder
Lateral Surface Area
of a Cylinder:
S  Ph
S  Ch
S  2 rh
• C = circumference of base
• r = radius of base
• h = height of the cylinder
Total Surface Area of a
Cylinder:
S  Ph  2B
2
S  2 rh  2 r
S  2 r  h  r 
Exercise 7
The net can be
folded to form
a cylinder.
What is the
approximate
lateral and
total surface
area of the
cylinder?
Height vs. Slant Height
By convention, h represents height and l
represents slant height.
Height vs. Slant Height
By convention, h represents height and l
represents slant height.
Exercise 8
Draw a net for the square pyramid below.
A  12  40 25
S  4  12  40 25
S  12  4  40 25
To find the lateral
surface area:
• Find the area of one triangle, then multiply by 4
Exercise 8
Draw a net for the square pyramid below.
A  12  40 25
S  4  12  40 25
S  12  4  40 25
S  12  4  s  l 
To find the lateral
surface area:
S  12 Pl
• Find the area of one triangle, then multiply by 4
Exercise 8
Draw a net for the square pyramid below.
A  12  40 25
S  4  12  40 25
S  12  4  40 25
S  12  4  s  l 
To find the total
surface area:
S  12 Pl
S  12 Pl  B
• Just add the area of the base to the lateral area
Surface Area of a Pyramid
Lateral Surface Area
of a Pyramid:
S  12 Pl
• P = perimeter of the base
• l = slant height of the
pyramid
Total Surface Area of a
Prism:
S  12 Pl  B
• B = area of the base
Exercise 9
Find the lateral and total surface area.
Exercise 10
You may have realized that the formula for the
lateral area for a prism and a cylinder are
basically the same. The same is true for the
formulas for a pyramid and a cone. Derive a
formula for the lateral area of a cone.
Lateral area of a Pyramid:
S  12 Pl
Lateral area of a Cone:
S  12 Cl
S  12  2 r  l
S   rl
Surface Area of a Cone
Lateral Surface Area
of a Cone:
S   rl
Total Surface Area of a
Cone:
S   rl   r
2
S   r l  r 
• r = radius of the base
• l = slant height of the
cone
Exercise 11
A traffic cone can be
approximated by a right
cone with radius 5.7
inches and height 18
inches. To the nearest
tenth of a square inch,
find the approximate
lateral area of the traffic
cone.
Tons of Formulas?
Really there’s just two formulas, one for
prisms/cylinders and one for pyramids/cones.
Assignment
• P. 806-9: 2-20
even, 21, 24, 25,
28, 30
• P. 814-7: 2, 3-21
odd, 22-25, 30
• Challenge
Problems: 3-5, 8, 9