1 standards 8, 10, 11 classifying solids problem 1 problem 2 surface area of cylinders volume of a...
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1
Standards 8, 10, 11
Classifying Solids
PROBLEM 1
PROBLEM 2
Surface Area of Cylinders
Volume of a Right Cylinder
PROBLEM 3
PROBLEM 4
PROBLEM 5
PROBLEM 6
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2
Standard 8:
Students know, derive, and solve problems involving perimeter, circumference, area, volume, lateral area, and surface area of common geometric figures.
Estándar 8:
Los estudiantes saben, derivan, y resuelven problemas involucrando perímetros, circunferencia, área, volumen, área lateral, y superficie de área de figuras geométricas comunes.
Standard 10:
Students compute areas of polygons including rectangles, scalene triangles, equilateral triangles, rhombi, parallelograms, and trapezoids.
Estándar 10:
Los estudiantes calculan áreas de polígonos incluyendo rectángulos, triángulos escalenos, triángulos equiláteros, rombos, paralelogramos, y trapezoides.
Standard 11:
Students determine how changes in dimensions affect the perimeter, area, and volume of common geomegtric figures and solids.
Estándar 11:
Los estudiantes determinan cambios en dimensiones que afectan perímetro, área, y volumen de figuras geométricas comunes y sólidos.
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PRISM PYRAMID
CYLINDER SPHERECONE
Standards 8, 10, 11SOLIDS
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Standards 8, 10, 11SURFACE AREA OF CYLINDERS
h
base
base
h
Lateral Area:
2 rL = h
2 rh
r
r
r
Total Surface Area = Lateral Area + 2(Base Area)
T= 2 rh + 2 r 2
r 2
r 2
h= heightr= radius
2 r
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VOLUME OF CYLINDERSStandards 8, 10, 11
h
r
r 2B=
V = Bh
V = r 2 h
RIGHT CYLINDER
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Find the lateral area, the surface area and volume of a right cylinder with a radius of 20 in and a height of 10 in.
Standards 8, 10, 11
10 in20 in
Lateral Area:
2 rL = h
L = 2 ( )( )10 in20 in
Total Surface Area = Lateral Area + 2(Base Area)
T= 2 rh + 2 r 2
T = 2 ( )( ) + 2 ( )210 in20 in 20 in
T= 400 in + 2(400 in )2 2
L=400 in2
T = 400 + 800in2 in2
T = 1200 in2
Volume:
V = r 2 h
V = ( )2( )10 in20 in
V= (400 in )(10 in) 2
V= 4000 in3
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Standards 8, 10, 11
Find the lateral area and the surface area of a cylinder with a circumference of 14 cm. and a height of 5cm.
C=2 r22
r= C2
r=2
r=7 cm
Finding the radius:
14
5 cm 7 cm
Lateral Area:
2 rL = h
L = 2 ( )( )5 cm7 cm
L= 70 cm 2
Total Surface Area = Lateral Area + 2(Base Area)
T= 2 rh + 2 r 2
T = 2 ( )( ) + 2 ( )25 cm7 cm 7 cm
T= 70 cm + 2(49 cm )22
T = 70 + 98 cm 2cm 2
T = 168 cm2
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Standards 8, 10, 11Find the Volume for the cylinder below:
25
First we find the height:
4
h
h
4 5
5 = 4 + h2 2 2
25 = 16 + h2
-16 -16
h = 92
h = 92
h = 3
Volume:
V = r 2 h
V = ( )2( )32
V= ( 4 )(3)
V= 12 unit3
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Standards 8, 10, 11
The surface area of a right cylinder is 400 cm. If the height is 12 cm., find the radius of the base.
Total Surface Area:
T= 2 rh + 2 r 2
h= 12 cm
Subtituting:
400 = 2(3.14)r(12) + 2(3.14)r2
=3.14
400 = 75.4 r + 6.28r2
-400 -400
0 = 6.28r + 75.4 r - 4002
We substitute values:
6.28
6.2875.4 75.4 -400
+ -X=-b b - 4ac
2a
2+_
where: 0 = aX +bX +c2
=-( ) ( ) - 4( )( )
2( )
2+_r
=-75.4 5685.16 + 10048
12.56
+_r
-75.4 15733.2 =
12.56
+_r
-75.4 125.43 =
12.56
+_r
-75.4+125.43 =
12.56r
12.5650.03
=r
4 cmr
12.56-200.83
=r
-16r
-75.4 -125.43 =
12.56r
Using the Quadratic Formula:
a= 6.28b= 75.4c= -400
From equation:
2
T= 400 cm2
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10
6
3
SIMILARITY IN SOLIDS
Standards 8, 10, 11
4
8
Are this two cylinders similar?
These cylinders are NOT SIMILAR
=4
683
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11
Standards 8, 10, 11
VOLUME 1 VOLUME 2IF THEN
AND r 1
r 2
=h1
h2
= 25
VOLUME 1 < VOLUME 2
V = r h 1 1 1
2 V = r h 2 2 22
Volume:
V = r 2 h
V r h
V r h=
1 1 1
2 2 2
2
2
=1 1 1
2 2 2
V r h
V r h
2
2
The ratio of the radii of two similar cylinders is 2:5. If the volume of the smaller cylinder is 40 units, what is the volume of the larger cylinder.3
V2
=2 25 5
402
V2
=4 2
25 540 40 8
V2 125=
(40)(125) = 8V2
8 8
V = 625 units23
=1 1 1
2 2 2
V r h
V r h
2
Substituting values:
THEN
AND IFThey are similar
What can you conclude about the ratio of the volumes and the ratio of the radii?
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