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SURFACE AREA and VOLUME
Solid Figures Introduction
PRISMS CYLINDERS
PYRAMIDS
CONESSPHERES
APPENDIX
SURFACE AREA and VOLUME
APPENDIX
SOLID FIGURES
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In a previous unit we discussed POLYGONS. POLYGONS are 2-dimensional objects.
That means they have length and width, but no depth.
Like they are cut out of paper.
SURFACE AREA and VOLUME
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SOLID FIGURES
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In this unit, we will learn about SOLIDS. A SOLID is a 3-dimensional (3D) object. That means it has length, width AND depth.
SURFACE AREA and VOLUME
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SOLID FIGURES
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When the SOLID is made up of POLYGONS, it is called a POLYHEDRON
Each POLYHEDRON is made up ofSurfaces (called faces)
Segments (called edges)
Corners (called vertices)
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APPENDIX
SOLID FIGURES
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There are many types of solids, we are going to study the 5 basic
types:PRISM PYRAMID
CYLINDER CONE SPHERE
Before we can do
anything with
SOLIDS, we must
understand the
difference between
these shapes
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SOLID FIGURES
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PRISMS: To create a prism, draw any polygon you like. Next, draw another polygon, just like the first one
BASE BASE BASE
Connect the corresponding vertices
SURFACE AREA and VOLUME
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SOLID FIGURES
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PRISMS:
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SOLID FIGURES
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PYRAMIDS: Pyramids have one baseInstead of connecting to an identical base…They connect to a point.
BASE BASE BASE
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SOLID FIGURES
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Pyramids:
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SOLID FIGURES
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When we name prisms and pyramids, we describe the base to tell what kind of prism or pyramid it is.
BASE PRISM PYRAMIDTriangle Triangular Prism Triangular PyramidRectangle Rectangular Prism Rectangular
PyramidPentagon Pentagonal Prism Pentagonal PyramidHexagon Hexagonal Prism Hexagonal PyramidHeptagon Heptangonal Prism Heptangonal
PyramidOctagon Octagonal Prism Octagonal PyramidDecagon Decagonal Prism Decagonal Pyramid
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SOLID FIGURES
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CYLINDERS: A cylinder is like a prism…But the bases are circles
SURFACE AREA and VOLUME
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SOLID FIGURES
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Cylinders:
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SOLID FIGURES
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CONES: A cone is a circular pyramid.
SURFACE AREA and VOLUME
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SOLID FIGURES
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Cones:
SURFACE AREA and VOLUME
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SOLID FIGURES
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SPHERES:
A sphere is a unique shape...It has no base or sides or edges or vertices.
SURFACE AREA and VOLUME
APPENDIX
SOLID FIGURES
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SPHERES:
SURFACE AREA and VOLUME
APPENDIX
SOLID FIGURES
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Identify each of the following shapes as a PRISM, PYRAMID, CYLINDER, CONE or SPHERE #1 #2 #3 #4
#5 #6
#7
#8 #9 #10
SPHERE CYLINDERPRISM
PYRAMID
CYLINDER CONE PYRAMID
PRISM SPHERE CONE & SPHERE
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APPENDIX
SOLID FIGURES
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So what are we going to do with these shapes?
We are going to calculate
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SURFACE AREA and VOLUME
APPENDIX
SOLID FIGURES
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SUFARCE AREA:Is literally the area of all the surfaces of a solid.
SURFACE AREA and VOLUME
APPENDIX
SOLID FIGURES
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SUFARCE AREA:Is literally the area of all the surfaces of a solid.
SURFACE AREA and VOLUME
APPENDIX
SOLID FIGURES
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SUFARCE AREA:Is literally the area of all the surfaces of a solid.
SURFACE AREA and VOLUME
APPENDIX
SOLID FIGURES
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SUFARCE AREA:Is literally the area of all the surfaces of a solid.
SURFACE AREA and VOLUME
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SOLID FIGURES
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VOLUME:Is the measure of how much “space” an object occupies.It is often thought of as how much water an object can hold.
SURFACE AREA and VOLUME
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SOLID FIGURES
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VOLUME:Or how many cubic units it would take to fill an object
This is why volume is measured in CUBIC UNITS
Like:
Cubic feet: ft3
Cubic centimeters cm3
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APPENDIX
PRISMS
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A prism is a solid with 2 congruent BASES connected by LATERAL FACES
Lateral face
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PRISMS
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To find the surface area of a PRISM, you can do it 2 ways:Method 1:
Find the area of each shape that makes up the Solid, and add them together
26
10
The front 2 x 10 = 20
20
The back 2 x 10 = 20The left 6 x 10 = 60The right 6 x 10 = 60The top 2 x 6 = 12The bottom 2 x 6 = 12
60
12
= 184
(in a rectangular prism any pair of opposite sides can be bases)
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APPENDIX
PRISMS
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To find the surface area of a PRISM, you can do it 2 ways:Method 2: Use the formula for SURFACE AREA:
26
10
BASE
•Identify the BASEs
•Find the Area and Perimeter of the base
•
B: area of the base P:
perimeter of the base
h: height
of the
prism
phBASAreaSurface
2.).(1016122 SA
16024184
12
16
10
P; 16A: 12
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APPENDIX
PRISMS
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A: 270m2
6m
Find the surface area of this regular decagonal prism:
4m
phBSA 2
P:60m
4602702 SA
Find the perimeter of the base6+6+6+6+6+6+6+6+6+6
=60
Area of the
basesLateral Area
2405402780m
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APPENDIX
PRISMS
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Finding Volume is easier!Find the VOLUME of this regular decagonal prism:
A: 270m2
6m4m
P:60m
hBVolume :
B: area of the base
h: height
of the
prism
4270Volume31080m
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APPENDIX
PRISMS
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RIGHT PRISMS vs OBLIQUE PRISMSA right prism is a prism where the lateral faces are all perpendicular to the bases.
An oblique prism is a prism where the lateral faces are NOT perpendicular to the bases.
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APPENDIX
PRISMS
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SUMMARY:Things you need to know about a prism:
A prism has 2 bases
All the lateral faces are rectangles
Surface Area = 2B + phB is the area of the basep is the perimeter of the
baseh is the height of the prismp x h gives the lateral area
Volume = B x hB is the area of the baseh is the height of the prism
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PRISMS
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Find the……Surface Area
…Lateral Area
…Volume
2cm
4cm 3cm
A:P:
6cm2
12cm
First we have to find that missing sideUse the Pythagorean theorem
4cm 3cm
5cm
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APPENDIX
PRISMS
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4cm 3cmFind the……Surface Area
…Lateral Area
…Volume
5cm
A:P:
6cm2
12cm
hB
phB 221262
2cm
2412236cm
224cm
26312cm
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APPENDIX
PRISMS
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Find the surface area and volume.
Area:Perimeter:
814
6
10
11224
136ft2
52ft
phBSA 25521362 SA
260272SA2532 ft
hBV
5136V3680 ft
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APPENDIX
PRISMS
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Find the volume.
81011
8149 1008
880
880100821888u
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PRISMS
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Archimedes problem.
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APPENDIX
CYLINDERS
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Since CYLINDERS are like prisms with circular bases,
We need to remember a few basic rules for circles.
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APPENDIX
CYLINDERS
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Find the Circumference and area of the circle:
12in6in
The red segment is the DIAMETER
rnceCircumfere 2
2rArea
We need the RADIUS
6212
in68.37
26
36204.113 in
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APPENDIX
CYLINDERS
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A CYLINDER is like a prism with circular basesThe surface area of a cylinder can be found with this formula
chBSA 2
h: height of
the cylinder
Most people find this formula easier: rhrSA 22 2
r
hArea of
the bases
Lateral Area
B: area of the base
C: circumference
of the base
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APPENDIX
CYLINDERS
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Find the SURFACE AREA of this cylinder:
ft3
ft10
rhrSA 22 2
103232 2 6018
78292.244 ft
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APPENDIX
CYLINDERS
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The VOLUME of a Cylinder:The VOLUME of a cylinder can be found with this formula
BhV
h: height of
the cylinder
The expanded version of the formula:
2V r h
r
h
B: area of the base
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APPENDIX
CYLINDERS
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Find the VOLUME of this cylinder:
ft3
ft10
2V r h23 10
9 10 90
3282.6f t
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APPENDIX
CYLINDERS
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SUMMARY
r
h2V r h
What you need to know about a cylinder:•A cylinder is a prism with circular bases
•The lateral area is a rectangle
•Surface Area:
•Volume:
•Lateral Area:
rhrSA 22 2
rhLA 2
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APPENDIX
CYLINDERS
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Find the Lateral Area, Surface Area and Volume:
11mm
9mm
2V r hrhrSA 22 2
rhLA 29112
198272.621 mm
9112112 2 198242
440 26.1381 mm
211 9 33419.46mm
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APPENDIX
CYLINDERS
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Find the surface area and volume of this shape 31ft
8ft
rhrSA 22 2 If it was a complete cylinder:
318282 2 236.1959 ft
But we only want half of it268.979 ft
What did we forget?
31ft16ft
49668.979 268.1475 ft
SURFACE AREA and VOLUME
APPENDIX
CYLINDERS
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Find the surface area and volume of this shape 31ft
8ft
If it was a complete cylinder: 2V r h
28 31 36,229.76f t
But we only want half of it33,114.88f t
SURFACE AREA and VOLUME
APPENDIX
PYRAMIDS
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A PYRAMID is a shape with a single base, where all the vertices of the base are connected to a single point (apex)
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APPENDIX
PYRAMIDS
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Just like with prisms, we will need to find:•Base Perimeter•Base Area•Height of the pyramid
However, with pyramids, we will also need to find a different measurement……SLANT HEIGHThe
ight
Slant height
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APPENDIX
PYRAMIDS
MENU
SLANT HEIGHT
Slant height is the distance someone would travel walking up the OUTSIDE of a pyramid.
SURFACE AREA and VOLUME
APPENDIX
PYRAMIDS
MENU
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APPENDIX
PYRAMIDS
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Finding the SURFACE AREA of a PYRAMID
PlBSA21
B: area of the base
P: perimeter of the base
L: Slant
height
Rectangular Pyramid
22m
22m
25m
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APPENDIX
PYRAMIDS
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Finding the SURFACE AREA of a PYRAMID
PlBSA21
Rectangular Pyramid
1484 88 252
Base Area:Base Perimeter:
484m2 88m
484 1100 Lateral Area21584m 22
m22m
25m
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APPENDIX
PYRAMIDS
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Finding the VOLUME of a PYRAMID
hBV 31
B: area of the base
h: height
Rectangular Pyramid
10ft 8ft
11ft1180
31
V
Base Area:Base Perimeter:
80ft2
36ft3293.3f t
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APPENDIX
PYRAMIDS
MENU
SUMMARY:Things you need to know about a pyramid:
A pyramid has only 1 base
All the lateral faces are triangles
Surface Area =B is the area of the basep is the perimeter of the basel is the slant height of the
pyramid½ x p x l is the lateral area
Volume =B is the area of the baseh is the height of the prism
PlBSA21
hBV 31
h
l
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APPENDIX
PYRAMIDS
MENU
Hey, I need to know the height. Hey, I need to know the slant height.
24
20
222 2012 h400144 2 h2562 h16h
16
15
222 158 L2289 LL17
Lateral Area
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APPENDIX
PYRAMIDS
MENU
Find the Lateral Area, Surface area and Volume:
PlBSA21
PlLA21
B:P:
40080
268021
21040u
SURFACE AREA and VOLUME
APPENDIX
PYRAMIDS
MENU
Find the Lateral Area, Surface area and Volume:
PlBSA21
B:P:
40080
1400 80 262SA
21440SA u
SURFACE AREA and VOLUME
APPENDIX
PYRAMIDS
MENU
Find the Lateral Area, Surface area and Volume:
B:P:
40080
13V B h
1 4003V h 10
h
2 2 210 26h 2100 676h 2 576h
24h
1 400 243V
33200V u
SURFACE AREA and VOLUME
APPENDIX
PYRAMIDS
MENU
Find the Volume:
264 48
111231
44
4448292u
SURFACE AREA and VOLUME
APPENDIX
CONES
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A CONE is basically a pyramid with a circular base
SURFACE AREA and VOLUME
APPENDIX
CONES
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R
HL
R: radius
H: height
L: slant height
lrBSA
hrV 2
31
Lateral Area
lrrSA
or
2
SURFACE AREA and VOLUME
APPENDIX
CONES
MENU
13
12i
n
5in
Find the slant height with the Pythagorean Theorem
Find the surface area and volume for the cone shown: 2SA r r l
13525 6525
90 2282.6in
213V r h
12531 2
100 3314in
SURFACE AREA and VOLUME
APPENDIX
CONES
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Find the surface area and volume:
hrV 2
31
lrrSA 2
15992 224.678 u
12931 2
208.3052 u
SURFACE AREA and VOLUME
APPENDIX
CONES
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Find the Lateral Area:
lrBSA
Lateral Area
5
5321.47 u
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APPENDIX
CONES
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Find the Volume:
Cylinder:
Cone:
TOTAL:hrV 22
18102 2 311304m
hrV 2
31
121031 2
31256m
3560,12 m
SURFACE AREA and VOLUME
APPENDIX
SPHERES
MENU
A sphere is just a ball.
It has 1 important measurement, that is used to find all its other propertiesThe RADIUS R
R
Official definition of a SPHERE:The collection of all points in space that
are the same distance (radius) from a point (center).
SURFACE AREA and VOLUME
APPENDIX
SPHERES
MENU
12m
3
34 rV
24 rSA 2124 576
264.1808 m
31234
2304
356.234,7 m
SURFACE AREA and VOLUME
APPENDIX
SPHERES
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Find the surface area:
24S r24 (8)
24 64 804 .in 8 in.
SURFACE AREA and VOLUME
APPENDIX
SPHERES
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Find the volume:
32 (3.14)3
343
V r
34 (2)3
4 (8)3
323
334 .ft
2 ft.
10in
SURFACE AREA and VOLUME
APPENDIX
SPHERES
MENU
Find the volume and surface area of the hemisphere:
24 rSA 2104
400
2rA 210
100
100200 3002942in
Half the sphere +the circle
33.2093 in
10in
SURFACE AREA and VOLUME
APPENDIX
SPHERES
MENU
Find the volume and surface area of the hemisphere:
3
34 rV 24 rSA
2942in
31034
37.4186 inWe only want half
SURFACE AREA and VOLUME
APPENDIX
SPHERES
MENU
Find the volume:
Cone:
Ice cream:
TOTAL:
21
34 3
rV
hrV 2
31 125
31 2
3314mm
215
34 3
367.261 mm
67.261314
67.575
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REVIEWS
ASSIGNMENTS
SURFACE AREA and VOLUME OPENERS
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