Geometry Chapter 10 – Spacial Reasoning
Lesson 1 – Solid Geometry
Learning Targets
LT10-1: Analyze 3-dimensional figures according
to their properties, nets, and cross-sections.
Success Criteria
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Classify 3-D figures.
Identify 3-D figures from a net.
Describe cross sections of 3-D figures.
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Face: A flat surface of the polygon.
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Edge: A segment that is the intersection of
two faces of the figure.
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Vertex: The point that is the intersection of
three or more faces of the figure.
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Cube: A prism with six square faces.
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Oblique: A figure whose axis is not perpendicular to the base.
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Right: A figure whose axis is perpendicular to its base.
Naming 3-Dimensional Figures: Pyramids and Prisms are named for the shape of their ____________.
Ex#1: Classify each 3-D figure. Name all vertices, edges, and bases.
A. Classify:
B. Classify:
vertices:
vertices:
edges:
edges:
bases:
bases:
C. Classify the figure below.
D. Classify the figure below.
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Net: A diagram of the faces of a three-dimensional figure arranged in such a way that the
diagram can be folded to form the three-dimensional figure.
Ex#2: Identify a 3-D figure from a net. Give as specific a name as possible.
A.
B.
C.
D.
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Cross section: The intersection of a three-dimensional figure and a plane.
Ex#3: Describe each cross section.
A.
B.
D.
C.
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 Targets
Success Criteria
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LT10-3: Apply the distance, midpoint, and
Euler's formulas to 3-D figures and
polyhedrons.
Use the Pythagorean Theorem in 3-D.
Graph figures in 3-D.
Find distances and midpoints in 3-D.
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Polyhedron: A closed three-dimensional figure formed by four or more polygons that
intersect only at their edges.
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Regular Polyhedron: A polyhedron in which all faces are congruent regular polygons and
the same number of faces meet at each vertex (also called a Platonic Solid).
Which of the following 3-D images is a polyhedra?
A.
B.
C.
E.
D.
F.
Ex: 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#1: Use the Pythagorean Theorem in 3-D.
A. Find the length of the diagonal of a 6cm by
8cm by 10cm rectangular prism.
•
B. Find the height of a rectangular prism with a
12in by 7in base and a 15in diagonal.
Space: The set of all points in three dimensions.
Ex#4: Graph each ordered triple.
A. (3, 2, 4)
B. (-2, -5, 3)
Ex#5: Graph each figure in 3-D.
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 given points in 3-D. 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: Find the distance between given points in 3-D.
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 Targets
LT10-4: Solve problems involving the
surface area of prisms and
cylinders.
Success Criteria
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•
•
•
Find lateral area (LA) and surface area (SA) of right
prisms.
Find lateral area (LA) and surface area (SA) of right
cylinders.
Find surface area of composite 3-D figures.
Describe effects of changing dimensions.
Use the diagrams below to label each of the following:
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•
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base
base edge
lateral face
lateral edge
lateral surface
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Lateral Area (LA): The sum of the areas of the lateral faces of a prism or pyramid,
or the area of the lateral surface of a cylinder or cone.
•
Surface Area (SA): The total area of all faces and curved surfaces of a threedimensional figure.
Prism
Cylinder
•
•
•
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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
Ex#1: Find the LA and SA of each right 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
Ex#2: Find the LA and SA for each right 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
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composite figure: A plane figure made up of triangles, rectangles, trapezoids,
circles, and other simple shapes, or a three-dimensional figure made up of prisms,
cones, pyramids, cylinders and other simple three-dimensional figures.
Ex#3: Find the SA of composite figures.
A.
B. What information would be needed to find the
SA of the hollow cylinder pictured below?
Ex#4: Describe effects of changing dimensions proportionally.
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 Targets
Success Criteria
LT10-5: Solve problems involving the
surface area of pyramids and cones.
•
•
•
•
Find lateral area (LA) and surface area (SA) of
pyramids.
Find lateral area (LA) and surface area (SA) of
right cones.
Describe effects of changing dimensions.
Find the surface area of composite 3-D figures.
Use the diagrams below to label each of the following:
•
•
vertex
slant height
•
•
lateral face
lateral surface
SA
ph
ph + 2B
Cylinder
2π r h
Pyramid
1
2
Sphere
l =
base
nonregular
pyramid
LA
Prism
Cone
•
•
lp
πrl
l p+B
πrl+π
•
regular
pyramid
axis
Volume
2π r h + 2 π r2
1
2
•
r2
Ex#1: Find the LA and SA for pyramids.
A. Find the LA and SA for a regular square
pyramid with base edge length 14 cm and
slant height 25 cm
B. Find the LA and SA for a regular
hexagonal pyramid with base edge 10 in and
slant height 16 in
Ex#2: Find the LA and SA of right cones.
A. Find the LA and SA of a right cone with
radius 9 cm and slant height 5 cm
B. Find the LA and SA of a right cone with
radius 8 in and altitude 15 in
Ex#3: Describe the 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.
Ex#4: Find the SA of 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 Targets
LT10-6: Solve problems involving
the volume of prisms and
cylinders.
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•
•
•
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Success Criteria
Find volume of prisms.
Find volume of cylinders.
Solve problems involving the volume of prisms and
cylinders.
Describe the effects of changing dimensions.
Find volume of 3-D composite figures.
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Volume: The number of non-overlapping unit
cubes of a given size that will exactly fill the
interior of a three-dimensional figure.
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Cavalieri's Principle: If two three-dimensional figures hae the same height and
have the same cross-sectional area at every level, they have the same volume.
LA
SA
Volume
ph
ph + 2B
Bh
2 π r h+ 2 π r 2
π r2 h
Prism
Cylinder
2π r h
Pyramid
1
2
Cone
lp
πrl
1
2
l p+B
πrl+π
r2
Sphere
Ex#1: Find the volume of each prism. Round to the nearest tenth, if necessary.
A.
B. a cube with edge length
C. the right regular hexagonal
15 in
prism below
Ex#2: Solve problems involving the volume of prisms and cylinders.
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.
Ex#3: 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 cm 2 and a
height equal to twice the radius
Ex#4: Describe effects of changing dimensions.
The radius and height of the cylinder below are multiplied by
volume.
Ex#5: Find the volume of composite 3-D figures.
A.
B.
2
. Describe the effect on the
3
Lesson 7 – Volume of Pyramids and Cones
Learning Targets
Success Criteria
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•
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LT10-7: Solve problems involving the
volume of pyramids and cones.
•
•
LA
SA
Volume
ph
ph + 2B
Bh
2π r h + 2 π r2
π r2 h
Prism
Cylinder
2π r h
Pyramid
1
2
Cone
Find volumes of pyramids.
Find volumes of cones.
Solve problems involving the volume of
pyramids and cones.
Describe effects of changing dimensions.
Find volumes of composite 3-D figures.
lp
πrl
1
2
l p+B
πrl+π
r2
1
3
1
3
Bh
π r2 h
Sphere
Ex#1: Find the volume of each pyramid.
A. A rectangular pyramid
B. The square pyramid with
with length 11 m, width 18 m base edge length 9 cm and
and height 23 m.
height 14 cm
C. The regular hexagonal
pyramid with height equal to
the apothem of the base.
Ex#2: Solve problems involving the volume of pyramids and cones.
An art gallery is a 6-story square pyramid with base area
1
2
acre
(1 acre = 4840 yd 2 , 1 story ≈ 10 ft.). Estimate the volume in cubic yards and in cubic feet.
Ex#3: Find the volume of a cone.
A. Find the volume of a cone B. Find the volume of a cone C. Find the volume of the
with radius 7 cm and height
with base circumference 25π cone pictured below.
15 cm
in and a height 2 in more than
twice the radius
Ex#4: Describe the effects of changing
dimensions. The diameter and height of the
cone are divided by 3. Describe the effect on
the volume.
Ex#5: Find the volume of the composite
figure. Round to the nearest tenth.
Lesson 8 – Spheres
Learning Targets
Success Criteria
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LT10-8: Solve problems involving the
surface area and volume of
spheres.
Solve problems involving the volume of spheres.
Solve problems involving the surface area of spheres.
Describe the effects of changing dimensions.
Find surface area and volume of composite figures.
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Sphere: The set of points in space that are a fixed
distance from a given point called the center of the
sphere.
•
great circle: A circle on a sphere that divides the
sphere into two hemispheres.
Prism
LA
SA
Volume
ph
ph + 2B
Bh
2 π r h+ 2 π r 2
π r2 h
Cylinder
2π r h
Pyramid
1
l
2
πrl
Cone
Sphere
p
1
l p+B
2
π r l + π r2
4 π r2
1
3
Bh
1
3
π r2 h
4
3
π r3
Ex#1: Solve problems involving the volume of spheres. Give answers in terms of π.
A. Find the volume of the
B. Find the diameter of a sphere C. Find the volume of the
sphere below.
hemisphere below.
with volume 36,000π cm 3 .
Ex#2: Solve problems involving the volume of spheres.
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?
Ex#3: Solve problems involving the SA of spheres. Give answers in terms of π.
A. Find the SA of a sphere with B. Find the volume of a sphere C. Find the SA of a sphere with
diameter of 76 cm
a great circle that has an area of
with a SA of 324π in 2
49π mi 2
Ex#4: Describe the effects of changing dimensions.
3
The radius of a sphere is multiplied by
. Describe the effect on the volume.
4
Chapter 10 – Spacial Reasoning
Lesson 10.1 P. 657 #13-28, 31-36, 38-44, 59
Lesson 10.3 P. 674 #15, 16, 18-22, 24, 25, 27, 28, 35-39, 45, 48
Lesson 10.4 P. 685 #14-16, 18-21, 23, 24, 26, 27, 32, 34, 38, 46, 48
Lesson 10.5 P. 694 #13-36 (skip #22, 25, 26, 27), 39, 40, 50
Lesson 10.6 P. 702 #13, 15, 16, 18, 19, 20, 23, 27, 30, 42
Lesson 10.7 P. 710 #14-17, 19, 20, 22, 26, 32, 37, 56
Lesson 10.8 P. 719 #13-22, 24, 27, 51