Geometry Unit 6.2
Three Dimensional Solids
8 May 2017
Agenda 5/8/2017
● Turn in homework - get set for the day. Bags out of the
way.
● Bulletin
● Wrap up from last time
○ Engineering Challenge - results
○ Challenge 2 - complete “net”
● Volume of prisms and cylinders
● Homework
● Final is Friday May 26th, forward planning
Homework. Will be collected on 5/8/2017 and graded
on neatness and completeness.
Volume and Surface Area of Rectangular Prisms
Instructions: Find the volume and surface area for each
rectangular prism.
● Show your work neatly and in a logical order. Use
separate paper if necessary.
● Pay attention to units.
On the line at the bottom write out the formula for volume of a
rectangular prism and a formula for its surface area.
Agenda 5/8/2017
● Turn in homework - get set for the day. Bags out of the
way. ✅
● Bulletin
● Wrap up from last time
○ Engineering Challenge - results
○ Challenge 2 - complete “net”
● Volume of prisms and cylinders
● Homework
● Final is Friday May 26th, forward planning
Engineering Challenge Part 2
Task: Design a rectangular box that holds more puffed rice
than your first container.
Do calculations and write out your plan before cutting this
time.
Aim is to maximize the volume of the container.
See results:
b) Cut a piece of colored paper up so that you
have square covers for each face of the cube.
How many squares do you need? 6
c) Rearrange the squares on the table top so they
form a rectangle. What is the area of this
rectangle?
c) Rearrange the squares on the table top so they
form a rectangle. What is the area of this
rectangle?
8 cm
12 cm
c) Rearrange the squares on the table top so they
form a rectangle. What is the area of this
rectangle?
4 cm
24 cm
c) Rearrange the squares on the table top so they
form a rectangle. What is the area of this
rectangle?
Area = lw
8 cm = (8cm)(12cm)
= 96 cm2
12 cm
d) How does the area we found in c) relate to the
cube we made in a)?
8 cm
4 cm
12 cm
4 cm
4 cm
d) How does the area we found in c) relate to the
cube we made in a)? The area is the surface area
of the cube.
8 cm
4 cm
12 cm
4 cm
4 cm
Volume and Surface Area - Challenge 2
Again put your 64 1cm3 cubes together to make
one large cube. Stand it on one of the squares we
cut.
Cube vs. rectangular prism
e) Now use the 6 squares to make a surface area
“jacket” for your cube - a net.
Tape the squares together so you can fold them
around the cube (create a net).
Don’t attach the last piece(s) of tape - so that you
can open the net and bring it up to the board.
Will there be only one design for the net?
Which of
these form a
net for a
cube?
✅
✅
✅
✅
Check up on terms:
The Prism (p. 15)
A prism is a polyhedron that consists of a
polygonal region and its translated image in a
parallel plane with the quadrilateral faces
connecting the corresponding edges.
That sounds pretty technical, and I guess it is.
Let’s try to clarify that definition with a few
examples.
The Prism (p. 15) cont.
The two congruent faces that have been
translated into parallel planes are called the
bases of the prism. The faces that are not bases
are called the lateral faces. All these are
examples of right prisms, which means the base
and lateral edges are perpendicular to one
another. The height is also a lateral edge.
How does this definition compare to what we might
have thought the base of a prism was before today?
Highlight the next paragraph.
A common misconception is that whatever face
the prism is “sitting” on is the base - that is not
how the base is determined. The height of the
prism is the perpendicular distance between the
two congruent bases.
We name a prism by its base shape.
Oblique Prisms (p. 16)
If the base and the lateral edges are not
perpendicular then the prism is called an oblique
prism. When the prism is oblique the lateral
faces are not rectangles, they are parallelograms
as shown.
Oblique Prisms (p. 16)
Prism Volume - the Stacking Principle (Cavalieri’s)
A powerful technique to use when calculating the
volume of a prism is stacking. For example, the
stack of CD’s is a solid made up of many square
caes upon each other or the volume of paper made
up of stacking many rectangular sheets up on each
other or money or coaster.
All of these cross sections have a height dimension
Prism Volume - the Stacking Principle
All of these cross sections have a height
dimension, but if we make that height infinitesimally
small we begin again to approximate the
relationship more accurately.
...So to calculate the volume of a prism we
calculate the area of the base and then multiply it
by the height of the prism...
VolumePrism = Bh, where B is the area of the base
and h is the height of the prism.
Just be careful and pay attention to the precise
meaning of the words in these mathematical
contexts.
Volumecube = Bh, where B is the area of the base and
h is the height of the prism.
V= B h
V = (area of square) (height)
5 cm
cube
V = (5cm x 5cm) (5cm)
V = 125cm3
Volumetriangular prism = Bh, where B is the area of the
base and h is the height of the prism.
V= B h
5
cm
V = (area of triangle) (height)
8
cm
10
cm
V = (½ 8cm x 5cm) (10cm)
V = 200cm3
Volumerectangular prism = Bh, where B is the area of the
base and h is the height of the prism.
3 cm
12 cm
V= B h
4cm
V = (area of rectangle)
(height)
V = (3cm x 4cm) (12cm)
V = 144cm3
Classwork. Find the volume of each prism. P. 18
4) Hexagonal prism.
6 cm
10cm
Area of a hexagon is ½ (apothem)(perimeter)
S = 6cm
a = 3√3cm
Perimeter
= 6cm x 6
= 36cm
10cm
Classwork. Find the volume of each prism. P. 18
4) Hexagonal prism.
Volume = B h
= (½ 3√3cmx36cm) (10cm)
=
6 cm
Classwork. Find the volume of each prism. P. 18
4) Hexagonal prism.
Volume = B h
= (½ 3√3cmx36cm) (10cm)
= 540√3cm3
6 cm
Shaye and Natalie’s Set of tables - Qu 5.
Asher and Artin’s set of tables - Qu 8
Lucine and Valeria’s set of tables - Qu 6
Alex and Gary’s set of tables Qu. 7
Solve - show work - explanations - come up to
share on white board.
G.GMD.3 Student Notes Lesson 5
The cylinder.
Not a prism - because of its circular base.
Does have 2 parallel identical bases (circles) and
a rectangular face.
(Think of creating a net for a cylinder.)
Possible nets for creating cylinders:
Height of cylinder?
Height of cylinder? Perpendicular distance between
2 circles (bases)
What will these lengths be the same as?
Circumference of circle 2π(radius)
Circumference of circle 2π(radius) (not to scale)
20π cm
G. GMD.3 Student N0tes Lesson 5 Add to the notes
page at the top
The Cylinder
Net for:
G. GMD.3 Student NOtes WS #3 Add to the notes
page
The Cylinder
Net for:
What is the length of the rectangle in the net? The
circumference of the circle that forms the base
base
base
base
base
Becomes curved
Lateral face
G. GMD.3 Student Notes WS #3 Add to the notes page
Cylinder Volume the Stacking Principle
CORRECTION NECESSARY IN BOX
Volume cylinder = Base x height = Bh = r2h
Volume cylinder = Base x height = Bh = r2h
Volume cylinder = Base x height = Bh = r2h
height
G. GMD.3 Student Notes Page 24
Volume cylinder = Base x height = Bh = r2h
Example #1
Height
5cm
r= 4cm
h= 5cm
V = Bh
V= ( r2)h
V= (4cm)25cm
V=80 cm3
G. GMD.3 Student Notes page 24
Volume cylinder = Base x height = Bh = B r2
Diameter = 6cm
Height
4cm
Example #2
Diameter = 6cm
r= 3cm
h= 4cm
V = Bh
V= ( r2)h
V= (3cm)24cm
V=36 cm3
G. GMD.3 Student Notes page 24
Volume cylinder = Base x height = Bh = r2h
What we need to
find is what?
Example #3
G. GMD.3 Student Notes page 24
Volume cylinder = Base x height = Bh = r2h
What we need to
find is:
Volume bottom
cylinder + volume
top cylinder Volume of the
cylindrical hole
through both
Example #3
V = Bh + Bh - Bh
V= (7cm)211cm +
(5cm)25cm - (1cm)216cm
V=648 cm3
Homework:
Lesson 4 p. 19 -20 Questions 1 -3 all parts except 2h.
1)
Shade the various possible bases now to make sense of
the problem
Lesson 5 p 25 - 27 All questions all parts.
Forward planning: Final is 8-10am Fri. May 26th
Wednesday May 10th Volume of Pyramids and Cones
Friday May 12th (Min day) Revolving 2-D shapes to get
volume (solids of revolution)
Tuesday May 16th Volume of spheres and review
Thursday May 18th Unit 6 Test
Monday May 22nd Review for final
Wednesday May 24th Review for final