8.8 ft 4 ft - Contoh Power Point Volume dan Luas Permukaan Bangun Ruang

CYLINDER
1

Standards 8, 10, 11

Classifying Solids
Surface Area of Cylinders
Volume of a Right Cylinder
PROBLEM 1
PROBLEM 2
PROBLEM 3
PROBLEM 4
PROBLEM 5
PROBLEM 6
2

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END SHOW

Standards 8, 10, 11

SOLIDS

PYRAMID

PRISM

CYLINDER

CONE
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SPHERE

3

SURFACE AREA OF CYLINDERS

Standards 8, 10, 11
r

base
r2

h

2

h

rh

r
2

r2

base

r

r
Lateral Area:
L =2 rh
Total Surface Area = Lateral Area + 2(Base Area)
T= 2 r h + 2 r 2
h= height
r= radius
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4

Standards 8, 10, 11

VOLUME OF CYLINDERS

r2

B=

V = Bh
V=

h

r2 h

r
RIGHT CYLINDER

5
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Standards 8, 10, 11
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.
Total Surface Area = Lateral Area + 2(Base Area)
rh + 2 r2

T= 2

T = 2 ( 20 in )( 10 in ) + 2 ( 20 in )

10 in
20 in

T= 400

2

in2 + 2(400 in2 )

T = 400 in 2 + 800 in 2
Lateral Area:
L =2

rh

L=2

(20 in)(10 in)

L=400

2

in

T = 1200 in 2
Volume:
V=
V=

r2 h
2
( 20 in ) ( 10 in )

V= (400 in2 )(10 in)
V= 4000

in3
6

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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.

Finding the radius:
C=2 r
2
2
r= C
2
r=

14
2

r=7 cm

5 cm

7 cm

Lateral Area:
L =2

rh

L=2

(7 cm)(5 cm)

L= 70

cm2

Total Surface Area = Lateral Area + 2(Base Area)

T= 2

rh + 2 r2

T = 2 ( 7 cm )( 5 cm ) + 2 ( 7 cm )
T= 70

2

cm2 + 2(49 cm 2 )

T = 70

cm 2 + 98

T = 168

cm 2

cm 2

7
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Standards 8, 10, 11

Find the Volume for the cylinder below:
h

2

4

5

First we find the height:
h

4

Volume:

2

2

5 = 4 + h2

5

25 = 16 + h2
-16 -16
2

h=9

V=
V=

r2 h
2
( 2) (3 )

V= ( 4 )(3)
V= 12

unit3

2

h= 9
h=3
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8

Standards 8, 10, 11
The surface area of a right cylinder is 400 cm.2 If the height is 12 cm., find the
radius of the base.
Total Surface Area:
Subtituting:
T= 2 r h + 2 r 2
400 = 2(3.14)r(12) + 2(3.14)r2
2
T= 400 cm
400 = 75.4 r + 6.28r 2
h= 12 cm
-400
-400
=3.14
0 = 6.28r 2 + 75.4 r - 400
Using the Quadratic Formula:

X=

_
-b +

2

b - 4ac
2a
2

where: 0 = aX +bX +c

From equation:

a= 6.28
b= 75.4
c= -400

We substitute values:
r=

-( 75.4 ) +
_

r=

( 75.4 )2 - 4(6.28 )(-400)
2( 6.28 )

_
-75.4 +

5685.16 + 10048
12.56

_
+
r = -75.4

15733.2
12.56
_
+
r = -75.4 125.43
12.56
+
-75.4 -125.43
-75.4+125.43
r
=
r=
12.56
12.56
50.03
-200.83
r=
r=
12.56
12.56
r

4 cm

r

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-16

9

Standards 8, 10, 11
SIMILARITY IN SOLIDS
Are this two cylinders similar?

4

6
3

8

4
6

= 8
3

These cylinders are NOT SIMILAR
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10

Standards 8, 10, 11
The ratio of the radii of two similar cylinders is 2:5. If the volume of the
smaller cylinder is 40 units,3 what is the volume of the larger cylinder.
VOLUME 1 < VOLUME 2
Volume:
V=

VOLUME 1

r2 h

V1
V2

=

IF

r12 h1
r 22 h 2

r12 h 1

V1 =

AND

VOLUME 2

V1
V2

=

V2 =

r1
r2

2

h1
h2

r22 h 2

V1

THEN

AND IF
They are similar

V2

r1
r2

=

=

h1
h2

r12 h1
r22 h2

= 2
5

Substituting values:
THEN

40 = 2
V2
5

2

2
5

40 = 4 2
V2 25 5

What can you conclude about the ratio of
the volumes and the ratio of the radii?

40
8
=
V2 125
(40)(125) = 8V2
8
8
V2 = 625 units 3

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11

CONE
12

Standards 8, 10, 11

Classifying Solids
Surface Area of Cones
Volume of a Right Cone
PROBLEM 1
PROBLEM 2
PROBLEM 3
PROBLEM 4
PROBLEM 5

13

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END SHOW

SURFACE AREA OF A RIGHT CIRCULAR CONE
perimeter of cone’s base
2 r
C=2 r

Standards 8, 10, 11
L= area of sector
Area of Circle
l

l

2

l

h

r
r
C=2 r

2
B= r

perimeter of cone’s base
area of sector
=
area of circle
perimeter of circle
area of sector
2

l

2

l
area of sector
2

=

l
L=area of sector =

TOTAL SURFACE AREA:

2 r

=

2 l
r
l
rl

l

C=2 l

2

T = area of sector + area of cone’s base
h= height
T=L+B
r = radius
T= r l + r2
l = slant height 14
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Standards 8, 10, 11
VOLUME OF A RIGHT CIRCULAR CONE

h

r
2

B= r

V=

1
Bh
3

V=

1
3

r2 h

h= height
r = radius

15
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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.
Lateral Area:

l

L=

rl

L=

(12 cm )(28.6 cm )

L = 1077.7 cm 2

h=26 cm

Calculating the base area:
12 cm = r
2

B= r
we need to find the slant height,
using the Pythagorean Theorem:
2

2

l = 26 + 12 2
2
l = 676 + 144
2

Calculating the volume:

2
B= r

B=

( 12 cm ) 2

B = 144
B= 452.2 cm 2
Calculating surface area:
T=L+B
2

l = 820

T = 1077.7 cm + 452.2 cm2

l

T = 1529.9 cm2

28.6 cm

1
3
1
V=
3
1
V=
3
V=

r2 h
2

( 12 cm ) (26 cm )
(144 cm2) (26 cm )

V = 3918.7 cm3

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16

Standards 8, 10, 11
Find the lateral area and the surface area and volume of a right cone whose
slant height is 9 ft and whose circumference at the base is 4 ft. Round your
answers to the nearest tenth.
we need to find the height, using
Lateral Area:
the Pythagorean Theorem:
C=4πft
L= r l
2
2
2
9 =h + 2
h
2
L= ( 2 ft )( 9.0 ft )
81 = h + 4
2 ft = r
-4
-4
L = 56.5 ft 2
l =9ft
h2 = 77
Calculating the base area:
2
h = 8.8 ft
We need to find the radius:
B= r



C=2 r
2
2
r= C
2
4
r=
2
r= 2 ft

B=

( 2 ft )

2

B= 4

T=L+B
T = 56.5 ft 2 + 12.6 ft 2

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1
3
1
V=
3
1
V=
3

V=

B= 12.6 ft 2
Calculating surface area:

T = 69.1 ft

Calculating the volume:

2

r2 h
( 2 ft

2

) ( 8.8 ft )

( 4 ft 2 ) ( 8.8 ft )

V = 36.8 ft 3

17

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.
Lateral Area:

l =22 m
r

h = 18 m

L=

rl

L=

( 13 m )( 22 m )

L = 898 m 2
Calculating the base area:
2
B= r

we need to find the radius, using
the Pythagorean Theorem:
2

22 = r 2 + 18
2

r = 13 m

( 13 m)

2

B = 169

2

484= r + 324
-324
-324
r2 = 160

B=

1
3
1
V=
3
1
V=
3

V=

B= 531 m 2
Calculating surface area:
T=L+B
T = 898 m2 + 531 m2
2

T = 1429 m
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Calculating the volume:
r2 h
2

( 13 m ) ( 18 m )
(169 m 2 ) ( 18m )

V = 3184 m 3

18

Standards 8, 10, 11
Find the lateral surface of a cone whose volume is 900 mm3 and whose radius is
15 mm. Round your answers to the closest tenth.
Now we draw the cone:

Calculating the height:
1
V=
3
1
( 900 ) =
3

r2 h

h =3.8
15= r

2

(15 ) h

(3) 900 = 1 ( 225 ) h
3
(3)
2700 = 225 h
225
225
h=

2700
706.5

h = 3.8 mm

Lateral Area:
L=

rl

L=

(15mm)(15.5 mm )

L

730 mm

2

we need to find the slant height,
using the Pythagorean Theorem:
2

2

l = 3.8 + 15 2
2
l = 14.4 + 225
2

l = 239.4
l

15.5 mm
19

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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,3 what is the approximate volume of the smaller cone?
VOLUME 1 > VOLUME 2
VOLUME 1

Volume:
1
V=
3
V1
V2

IF
2

r h

=

r12 h1
r 22

h2

V1 =

1
3

r12 h 1

V1

AND

VOLUME 2
1 r2 h
V2 =
2 2
3

V2

=

r1
r2

2

h1
h2

THEN

AND IF
They are similar

1
V2
3
=
V1 1
3
r1
r2

=

h1
h2

r12 h 1
r22 h 2

= 8
3

Substituting values:
THEN 2090 = 8
V2
3

2

8
3

2090= 64 8
V2 9 3

What can you conclude about the ratio
of the volumes and the ratio of the
radii?

2090 512
=
V2 27

(27)(2090) = 512V2
512
512
V2 110 units 3

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20