9.1
Name ______________________________ Per.______
Exploring Solids
VOCABULARY
SOLID:
Prism:
POLYHEDRON
• Face
• Vertex
• Edge
a solid with 2 congruent bases that lie in parallel planes.
faces are rectangles)
Pyramid:
a solid whose base is a
polygon and lateral faces
are triangles.
Cone:
(lateral
a solid with a circular base
with a vertex not on the same
plane.
BASES vs. LATERAL FACES
• Cylinder or Cone: The bases area the circular parts
Cylinder:
Sphere:
a solid with two
parallel circular bases.
set of all points in space
that is equidistant from a
given point.
Ex: Shade in the base(s) and label/draw the
height with an h
• Prism: The bases are the ______________ and ____________ faces
(lateral faces are rectangles!)
• Pyramid: The base is the non-congruent face that is non-coplanar to
the vertex.
HEIGHT: The distance from base to base OR base to vertex.
* To Find it: Highlight to bases, it’s the # connecting the 2 bases at a
right ∡
\
Example 1: A) Name the solid.
solid.
B) Shade its base(s). C) Label an h for the height of the
1.
2.
3.
4.
5.
6.
Example 2:
(P).
Name the solid, then find the area of the base (B) and perimeter of the base
1
1. (a) Solid Name:___________________
(b) Shade in the Base(s). What Shape is the Base?_______________________
(c) What is the height of the solid? _______
(d) Find the Area of the Base and the Perimeter of the Base
B= ______________________
P = ______________________
2. (a) Solid Name:___________________
(b) Shade in the Base(s). What Shape is the Base?_______________________
(c) What is the height of the solid? _______
(d) Find the Area of the Base and the Perimeter of the Base
B = ______________________
P = ______________________
3. (a) Solid Name:___________________
(b) Shade in the Base(s). What Shape is the Base?_______________________
(c) What is the height of the solid? _______
(d) Find the Area of the Base and the Perimeter of the Base
B = ______________________
P = ______________________
4. (a) Solid Name:___________________
(b) Shade in the Base(s). What Shape is the Base?_______________________
(c) What is the height of the solid? _______
20
(Hint! Its not one of the given #’s)
(d) Find the Area of the Base and the Perimeter of the Base
B = ______________________
P = ______________________
2
5. (a) Solid Name:___________________
(b) Shade in the Base(s). What Shape is the Base?_______________________
(c) What is the height of the solid? _______
(d) Find the Area of the Base and the Perimeter of the Base
B = ______________________
P = ______________________
6. (a) Solid Name:___________________
(b) Shade in the Base(s). What Shape is the Base?_______________________
(c) What is the height of the solid? _______
14
(d) Find the Area of the Base and the Perimeter of the Base
B = ______________________
P = ______________________
7. (a) Solid Name:___________________
(b) Shade in the Base(s). What Shape is the Base?_______________________
(c) What is the height of the solid? _______
(d) Find the Area of the Base and the Perimeter of the Base
9m
B = ______________________
P = ______________________
3
9.2
Height vs. Slant Height & Simple Volume
Slant height (ℓ)
ℓ
• Which 2 solids have
a slant height?
Example 1: Find the indicated measure.
1. Given the diameter of the cone is 24m, find the slant
height, ℓ.
16 m
* TAKE a LOOK!*
Notice that the height(h) and
slant height (ℓ) make a
_______ _____________
ℓ
2. Given height of the pyramid is 12ft and the slant height
is 13 ft, find the area of the base of the pyramid.
ℓ
3. Find the height of the pyramid shown below.
4. If the height of the cone is 5√3 cm, then find…
30°
(a) slant height
(b) area of the base (B)
Example 2: Finding the Volume of a Box
1. Find the volume of the box below.
2. If the volume of the box below is 192 cubic inches, then
find the width of the box.
3. Find the total volume of the shape shown to the right if all edges meet at right angles.
3 cm
2 cm
4 cm
10 cm
Example 3: Box nets
1. Find the volume of the figure if it is made into an open
box (no lid).
2 cm
2. Find the volume of the figure if it is made into an open
box (no lid).
4
9.3
Surface Area & Volume
SURFACE AREA FORMULAS
VOLUME FORMULAS
the ______ of its lateral faces
The amount of ____-dimensional space a
figure occupies.
SA = 2B + Ph
V = Bh
PRISM
PYRAMID
SA = B +
CYLINDER
CONE
SPHERE+
Pℓ
V=
2
SA = 2𝜋r 2 + 2𝜋rh
V=
SA = 4𝜋r 2
V=
• P → ______________ of the base
- Use the formula for the shape of the base
- add all the sides!
Example 1: Name the solid then find the indicated measure.
1. Volume
2. Surface area
3. Volume
5. Volume
4. Surface area
6. Surface area
5
3
V = 𝜋r 2 h
SA = 𝜋r 2 + 𝜋rℓ
• B → ________ of the base
Bh
𝜋r2 h
3
4𝜋r3
3
7. Volume
8. Volume (round to the nearest tenth)
26cm
9. Surface Area
10. Surface area
11. Surface area
12. Surface area
13. Surface area
14. Volume
6
15. Find the total volume of the solid.
16. Find the total volume of the solid.
24
8
8
Spiral Review:
1. Which is a possible third side length of a triangle with
side lengths 7 and 24?
2. Find the measure of angle 1.
A. 42°
B. 48°
C. 58°
D. 74°
A. 14
B. 17
C. 23
D. 34
1
3. Which of the following best describes the triangles
shown below?
14
71°
4. A ladder broke into two pieces creating a right angle
where it broke. Find the original height of the ladder.
42°
16
A. 9
B. 15
C. 23
D. 40
67°
42°
8 ft
17 ft
A. both similar and congruent
B. congruent but not similar
C. similar but not congruent
D. neither similar nor congruent
6. ̅̅̅̅
𝑂𝑃 is tangent to ⊙𝑁. If 𝑁𝑃 = 10 and 𝑂𝑃 = 8, what is
the circumference of ⊙𝑁?
5. What is the area in square units of the trapezoid shown
below?
A.
B.
C.
D.
160
80
396
76
(0, 8)
(9, 8)
P
N
O
(11, 0)
7
9.4
Working Backwards
Given the surface area/volume and Asked to find a missing Length:
1) Identify which formula you need to use
2) Plug in what you have
Example 1: Solve for the missing length.
1. Find the value of x if the surface area of the figure is 200 ft2.
3. Find the value of x if the volume of the figure
is 284.1 mm3
2. The volume of a right cylinder is 3600𝜋
cubic centimeters and the height is 16
centimeters. Find the radius.
4. Find the missing measurements of the cone if its volume is 75𝜋
ft3.
ℓ
5. The volume of a sphere is 288𝜋 cubic
meters. What is the length of its radius.
6. The regular pyramid below with a square base has a surface area of
1071 square meters. Find the length of its slant height.
ℓ
17 m
8
7. Jessica wants to paint her basketball
6. Find the height (not the slant height!) of the pyramid shown below if its
orange and needs enough paint to
surface area is 800 ft2.
cover all 100𝜋 square centimeters of it.
What is the length of its radius?
MIXED REVIEW
1. Find the surface area of the figure below.
2. Find the volume of the figure below.
3. Find the volume of the figure below.
4. Find the surface area of the figure below.
13in
10in
10in
5. Find the volume of the figure below.
6. Find the surface area of the figure below.
9
9.5
Similar Solids
Vocabulary
Similar solids
SIMILAR SOLIDS THEOREM
SIDE2 : SIDE2 = AREA : AREA
SIDE3 : SIDE3 = VOLUME:VOLUME
(𝑠𝑎 )2 𝐴𝑎
=
(𝑠𝑏 )2 𝐴𝑏
(𝑠𝑎 )3 𝑉𝑎
=
(𝑠𝑏 )3 𝑉𝑏
Example 1: Use the scale factor (side: side) of similar solids
1. If the similarity ratio between to cylinders is 3:7, then
2. The ratio of similarity of two cones is 4:3. If the volume
find….
of the larger cone is 256𝜋 cm3, what is the volume of the
(a) the surface area ratio
smaller cone?
(b) the volume ratio
3. Two similar regular pyramids have a surface
area ratio of 5:11. If the surface area of the
smaller regular pyramid is 100 square feet,
then what is the surface area of the larger
regular pyramid?
4. Cylinders A and B are similar with a similarity ratio of 2:5. Find
the surface area and volume of Cylinder B given that the surface
area of Cylinder A is 96 square feet and the volume of a Cylinder
A is 128 cubic feet.
5. The surface area ratio of two similar spheres is 25:36. If
the volume of the larger sphere is 648𝜋 cubic meters,
then what is the volume of the smaller sphere?
6. The volume ratio of two prisms is 8:27. If the surface
area of the smaller prism is 20 square inches, then what
is the surface area of the larger prism?
10
7. Find the surface area and volume of the similar solids.
SA = ______
SA = ______
V = ______
V = ______
MIXED PRACTICE
1. Find the volume of the figure below.
2. Find the surface area of the figure below.
3. The ratio of similarity of two hexagonal prisms is 5:9. If
the volume of the larger prism is 15309 m3, what is the
volume of the smaller cone?
4. Find the volume of the figure below.
5. The shape below has edges that meet at right angles.
What is the total volume.
6. Find the slant height of the figure below if its surface
area is 56 ft2.
4m
ℓ
2m
8m
I
J
12 m
8m
6m
5m
11
7. Find the surface area of the right cone with a radius of
12cm and a height of 35cm.
8. Find the surface area of the figure below.
12cm
35cm
9. If a right cylinder with a height of 17 meters can hold
612𝜋 cubic meters of water, then what is the length of its
diameter?
10. The volume ratio of two hexagonal prisms is 8:343
and the surface area of the smaller hexagonal prism is
76 square units. Find the surface area of the larger
hexagonal prism.
Review for Test
1. Find the volume of the regular pyramid.
2. Find the surface area.
3. Find the volume and surface area of the figure.
4. A large beach ball can hold 7776𝜋
cubic centimeters of air. What is the
radius of the ball?
5. Find the surface area of a right cone.
16
12
6. Find the volume of the figure if it is made into
an open box (no lid).
7. Find the volume of the given figure.
8. Find the volume of the solid made up of a cone and a hemisphere.
10. The volume ratio of two prisms is 125:64. If the
surface area of the smaller prism is 112 square inches,
then what is the surface area of the larger prism?
9. Find the volume of a sphere given the surface
area is 484𝜋 square inches.
11. The ratio of similarity of two boxes is 3:11. If the larger
box has a surface area of 968 square inches, then what is
the surface area of the smaller box?
(b) What is the volume ratio between the two boxes?
12. Given the volume of the pyramid is 400 ft3, what is the length of the
slant height?
13. Find the volume of the figure shown
below if all lines meet at right angles.
5m
ℓ
7m
3m
9m
13
6m
14