Geometry Chapter 10 – Spacial Reasoning
Lesson 1 – Solid Geometry
Learning Target (LT-1) Analyze 3-dimensional figures according to their properties, nets, and
cross-sections.
•
face
•
edge
•
vertex
•
cube
Naming 3-Dimensional Figures: Pyramids and Prisms are named for the shape of their ____________.
Ex1: Classify each figure. Name all vertices, edges, and bases.
A.
B.
C.
D. Classify a soup can.
•
Net
Ex2: Describe the three-dimensional figure that can be made from the given nets.
A.
B.
C.
D.
•
Cross section
Ex3: Describe each cross section.
A.
B.
C.
D.
E. A piece of cheese is in the shape of an equilateral triangular
prism. How can you slice the cheese to make each shape?
i. Equilateral Triangle?
ii. Rectangle?
iii. Isosceles Triangle?
Lesson 3 – Formulas in Three-Dimensions
Learning Target (LT-3) Apply the distance, midpoint, and Euler's formulas to 3-dimensional
figures and polyhedrons.
•
Polyhedron
•
Regular Polyhedron
Ex1: Which of the following 3-D images is a polyhedra?
A.
B.
C.
E.
F.
D.
Ex 2: Find the number of vertices, edges, and faces of each polyhedron. Use your results to verify
Euler's Formula.
A.
B.
V:
V:
E:
E:
F:
F:
Ex 3: Find the unknown dimension in each figure.
A. the length of the diagonal of a 6 cm by 8 cm B. the height of a rectangular prism with a 12 in
by 10 cm rectangular prism.
by 7 in base and a 15 in diagonal.
•
Space
Ex 4: Graph each ordered triple.
A. (3, 2, 4)
B. (-2, -5, 3)
Ex 5: Graph each figure.
A. A rectangular prism with length 5, width 3,
height 4, and one vertex at (0, 0, 0).
B. A cone with radius 3, height 5, and the base
centered at (0, 0, 0).
Ex 6: Find the distance between the given points. Find the midpoint of the segment with the given
endpoints. Round to the nearest tenth, if necessary.
A. (0, 0, 0) and (2, 8, 5)
B. (6, 11, 3) and (4, 6, 12)
Ex 7: Trevor drove 12 miles east and 25 miles south from a cabin while gaining 0.1 miles in elevation.
Samira drove 8 miles west and 17 miles north from the cabin while gaining 0.15 miles in elevation.
How far apart were the drivers?
Lesson 4 – Surface Area of Prisms and Cylinders
Learning Target (LT-4) Solve problems involving the surface area of prisms and cylinders.
Use the diagrams below to label each of the following:
•
•
•
•
•
base
base edge
lateral face
lateral edge
lateral surface
•
Lateral Area (LA):
•
Surface Area (SA):
Prism
Cylinder
•
•
•
•
right prism
oblique prism
altitude
axis
LA
SA
ph
ph + 2B
2π r h + 2 π r2
2π r h
Pyramid
Cone
Sphere
p=
B=
h=
r=
Volume
Ex1: Find the LA and SA of each prism below. Round to the nearest tenth, if necessary.
A.
B. a right regular triangular prism with height
20 cm and base edge of length 10 cm
Ex2: Find the LA and SA for each cylinder below. Give your answer in terms of π.
A.
B. a cylinder with circumference 24π cm and
a height equal to half the radius
•
composite figure
Ex3: Find the SA of the composite figures.
A.
B. What information would be needed to find the
SA of the hollow cylinder pictured below?
Ex4: Exploring effects of changing dimensions.
A. The edge of a cube is tripled. Describe the
effect on the surface area.
B. The height and diameter of the cylinder are
1
multiplied by
. Describe the effect on the
2
surface area.
Lesson 5 – Surface Area of Pyramids and Cones
Learning Target (LT-5) Solve problems involving the surface area of pyramids and cones.
Use the diagrams below to label each of the following:
•
•
•
•
vertex
slant height
lateral face
lateral surface
•
•
•
•
base
regular pyramid
nonregular pyramid
axis
Prism
LA
SA
ph
ph + 2B
Cylinder
2π r h
2π r h + 2 π r2
Pyramid
1
lp
2
πrl
1
lp + B
2
π r l + π r2
Cone
Volume
Sphere
l=
Ex1: Find the LA and SA for the pyramids below.
A. a regular square pyramid with base edge
B. a regular hexagonal pyramid with base
length 14 cm and slant height 25 cm
edge 10 in and slant height 16 in
Ex2: Find the LA and SA of each cone.
A. a right cone with radius 9 cm and slant
height 5 cm
B. a right cone with radius 8 in and altitude
15 in
Ex3: Exploring effects of changing dimensions.
The base edge length and slant height of the regular hexagonal pyramid are both divided by 5.
Describe the effect on the surface area.
Ex4: Composite 3-Dimensional figures.
A. Find the SA for the figure below.
B. If the pattern below is used to make a
paper cup, what is the diameter of the cup?
Lesson 6 – Volume of Prisms and Cylinders
Learning Target (LT-6) Solve problems involving the volume of prisms and cylinders.
•
Volume
•
Cavalieri's Principle
Prism
LA
SA
Volume
ph
ph + 2B
Bh
π r2 h
Cylinder
2π r h
2 π r h+ 2 π r 2
Pyramid
1
lp
2
πrl
1
lp + B
2
π r l + π r2
Cone
Sphere
Ex1: Find the volume of each prism. Round to the nearest tenth, if necessary.
A.
B. a cube with edge length
15 in
C. the right regular hexagonal
prism below
Ex2: A swimming pool is a rectangular prism. Estimate the volume of water in the pool in
gallons when it is completely full. (Hint: 1 gallon is about 0.134 ft 3 ). The density of
water is about 8.33 pounds per gallon. Estimate the weight of the water in pounds.
Ex3: Find the volume of each cylinder. Give your answer in terms of π and rounded to the
nearest tenth.
A.
B. a cylinder with base area 121π cm2 and a
height equal to twice the radius
Ex4: Exploring effects of changing dimensions.
The radius and height of the cylinder below are multiplied by
volume.
Ex5: Find the volume of the composite 3-D figures below.
A.
B.
2
. Describe the effect on the
3
Lesson 7 – Volume of Pyramids and Cones
Learning Target (LT-7) Solve problems involving the volume of pyramids and cones.
LA
SA
Volume
ph + 2B
Prism
Cylinder
2π r h
Pyramid
1
lp
2
πrl
Cone
Bh
ph
2π r h + 2 π r
2
π r2 h
1
lp + B
2
π r l + π r2
1
3
1
3
Bh
π r2 h
Sphere
Ex1: Find the volume of each pyramid.
A. a rectangular pyramid with B. the square pyramid with
length 11 m, width 18 m and base edge length 9 cm and
height 23 m.
height 14 cm
Ex2: An art gallery is a 6-story square pyramid with base area
C. the regular hexagonal
pyramid with height equal to
the apothem of the base.
1
2
acre (1 acre = 4840 yd 2 ,
1 story ≈ 10 ft.) Estimate the volume in cubic yards and in cubic feet.
Ex3: Find the volume of each cone.
A. a cone with radius 7 cm
and height 15 cm
B. a cone with base
circumference 25π in and a
height 2 in more than twice
the radius
Ex4: Exploring the effects of changing
dimensions. The diameter and height of the
cone are divided by 3. Describe the effect on
the volume.
C.
Ex5: Composite figures. Find the volume of
the composite figure. Round to the nearest
tenth.
Lesson 8 – Spheres
Learning Target (LT-8) Solve problems involving the surface area and volume of spheres
•
sphere
•
great circle
Prism
LA
SA
Volume
ph
ph + 2B
Bh
π r2 h
Cylinder
2π r h
2 π r h+ 2 π r 2
Pyramid
1
lp
2
πrl
1
lp + B
2
π r l + π r2
Cone
Sphere
4 π r2
1
3
Bh
1
3
π r2 h
4
3
π r3
Ex1: Find the indicated measurement of each sphere. Give answers in terms of π.
A. the volume of the sphere
below
B. the diameter of a sphere with C. the volume of the hemisphere
below
volume 36,000π cm 3
Ex2: A sporting goods store sells exercise balls in two sizes, standard (22 in diameter) and jumbo (34
in diameter). How many times as great is the volume of a jumbo ball as the volume of a standard ball?
Ex3: Find the indicated measurements of the spheres below. Give answers in terms of π.
A. the SA of a sphere with
diameter of 76 cm
B. the volume of a sphere with a C. the SA of a sphere with a
great circle that has an area of
SA of 324π in 2
49π mi 2
Ex4: Exploring the effects of changing dimensions.
The radius of a sphere is multiplied by
3
. Describe the effect on the volume.
4