6 3 surf_area_vol_cones
TRANSCRIPT
1
Standards 8, 10, 11
Classifying Solids
PROBLEM 1
PROBLEM 2
Surface Area of Cones
Volume of a Right Cone
PROBLEM 3
PROBLEM 4
PROBLEM 5
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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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3
PRISM PYRAMID
CYLINDER SPHERECONE
Standards 8, 10, 11SOLIDS
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4
h
r
ll
2 r
r
2 rL= area of sector
2
l
area of sectorarea of circle
Perimeter of sector / perimeter of cone’s baseperimeter of circle=
2C= l
2 l=
area of sector
2 rC=
Area of Circle2
l
2 rC=
perimeter of cone’s base
r2
l l=
area of sector 2
l2
l
TOTAL SURFACE AREA:
T = area of sector + area of cone’s base
2rB=
area of sector lr=L=
T = L + B
T= 2rlr +
SURFACE AREA OF A RIGHT CIRCULAR CONE /Curve Surface (Luas selimut Kerucut)
Standards 8, 10, 11
h= height
r = radius
l = slant heightLateral Area PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
5
h
r
VOLUME OF A RIGHT CIRCULAR CONEStandards 8, 10, 11
2rB=
V = Bh13
V = 2r1
3h
h= height
r = radius
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6
Standards 8, 10, 11
Find the lateral area, the surface area and volume of a right cone with a height of 26 cm and a radius of 12 cm. Round your answers to the nearest tenth.
h
r
=26 cm
12 cm =
Lateral Area:
lrL=
we need to find the slant height, using the Pythagorean Theorem:
l
l = 26 + 122 22
l = 676 + 1442
l = 8202
l 28.6 cm
Calculating the base area:
2rB=
2rB=
12 cmB= ( )2
B= 452.2 cm 2
Calculating surface area:
T = L + B
L= ( )( )12 cm 28.6 cm
L = 1077.7 cm2
T = 1077.7 cm + 452.2 cm2 2
T = 1529.9 cm2
Calculating the volume:
2rV = 13
h
26 cmV = 13
( ) ( )2
12 cm
V = 3918.7 cm3
B = 144
V = 13
( ) ( )26 cm144 cm2
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7
Standards 8, 10, 11
Find the lateral area and the surface area and volume of a right cone whose slant height is 9 m and whose circumference at the base is 4 m. Round your answers to the nearest tenth.
9 m
h
r l
We need to find the radius:
C=2 r22
r= C2
r=2
r= 2 m
4
2 ft =
we need to find the height, using the Pythagorean Theorem:
C=4πft9 = h + 22 2 2
81 = h + 42
-4 -4
h = 772
h = 8.8 m
Lateral Area:
lrL=
L= ( )( ) 2 m 9.0 m
L = 56.5 m2
Calculating the base area:2rB=
2 B= ( )2
B= 12.6 m2
B = 4
Calculating surface area:
T = L + B
T = 56.5 m + 12.6 m22
T = 69.1 m2
Calculating the volume:
2rV = 13
h
8.8 mV = 13
( ) ( 2
2 m
V = 36.8 m3
V = 13
( ) ( ) 8.8 m 4 m2
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8
Standards 8, 10, 11
Find the lateral area, the surface area, and the volume of a right cone whose height is 18 m and whose slant height is 22 m. Round your answers to the nearest unit.
h = 18 mr
l =22 m
we need to find the radius, using the Pythagorean Theorem:
22 = r + 182 2 2
484= r + 3242
-324 -324
r = 1602
r = 13 m
Lateral Area:
lrL=
L= ( )( ) 13 m 22 m
L = 898 m2
Calculating the base area:2rB=
13 mB= ( )2
B= 531 m 2
B = 169
Calculating surface area:
T = L + B
T = 898 m + 531 m22
T = 1429 m2
Calculating the volume:
2rV = 13
h
18 mV = 13
( ) ( )2
13 m
V = 3184 m3
V = 13
( ) ( ) 18m 169 m2
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9
Standards 8, 10, 11
Find the lateral surface of a cone whose volume is 900 mm and whose radius is 15 mm. Round your answers to the closest tenth.
3
2rV = 13
h
( ) = 13
( ) h2
900 15
900 = 13
( 225 ) h(3) (3)
2700 = 225 h 225 225
h= 2700706.5
h = 3.8 mm
Now we draw the cone:
h
r15=
we need to find the slant height, using the Pythagorean Theorem:
l = 3.8 + 152 22
l = 14.4 + 2252
l = 239.42
l 15.5 mm
Lateral Area:
lrL=
L= ( )( ) 15mm 15.5 mm
L 730 mm2
=3.8
Calculating the height:
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10
Standards 8, 10, 11
The ratio of the radii of two similar cones is 3:8. If the volume of the larger cone is 2090 units, what is the approximate volume of the smaller cone? 3
VOLUME 1 > VOLUME 2
2rV = 13
h
Volume:VOLUME 1 VOLUME 2
IFV = h2r1
31 1 1 V = h2r132 2 2
THENV1
h2r13 1 1V2
h2r13 2 2
=
AND r 1
r 2
=h1
h2
= 83
V r h
V r h=
1 1 1
2 2 2
2
2
V2
=8 83 3
20902
V2
=64 89 3
2090 2090 512V2 27
=
(27)(2090) = 512V2
512 512
=1 1 1
2 2 2
V r h
V r h
2
Substituting values:
THEN
AND IFThey are similar
V 110 units23
What can you conclude about the ratio of the volumes and the ratio of the radii?
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