Embed Size (px)
Citation preview
Geometry
12.2 – 12.5 Prep for the STAR Test
Geometry
12.2 Pyramids
The lateral area of a regular pyramid with n lateral faces is n times the area of one lateral face.
6
10
7) Find the lateral area and total area of this regular pyramid.
Let’s solve #7 as we learn the formula for lateral area!!
The best way to know the formulas is to understand themrather than memorize them.L.A. = nF
6
10
A = ½ b(h)A = 3(10)A = 30
LA = nFLA = 6(30) LA = 180 square units
OR…
The lateral area of a regular pyramid equals half the perimeter of the base times the slant height.
L.A. = ½ plWhy? Imagine a curtain on the ground around the pyramid(perimeter) and you are pulling the curtain up the slant height. Since the curtain will wrap around triangles we need the ½ .
LA = ½ plLA = ½ 36(10)
LA = 180 square unitsWe have 6 triangles!
7) Find the lateral area and total area of this regular pyramid.
The total area of a pyramid is its lateral area plus the area of its base.
6
10
T.A. = L.A. + B That makes sense!
A = ½ a(p)
6
30
3
3√3
A = ½ 3√3(36)A = 3√3(18)
A = 54√3
TA = LA + B
TA = 180 + 54√3 sq. units
Let’s solve #9 as we learn the formula volume!!
9. Find the volume of a regular square pyramid with base edge 24 and lateral edge 24.
Draw a square pyramid with the given dimensions.
24
The volume of a pyramid equals one third the area of the base times the height of the pyramid.
13
V BhWhy? Because a rectangular prism with the same baseand same height would be V = bh and the pyramid, believeit or not, would hold 1/3 the amount of water.
h l
1/3 the volume of its prism.
V = 1/3 B(h)
V = 1/3 24(24)(h)
V = 8(24)(h)
V = 192(h)12
24
Must be a 30-60-90.
12√312√3
12
122 + x2 = (12√3)2
12√2
V = 192(12√2)V = 2304√2 sq. units
12.3Cylinders and Cones
To find lateral area (L.A.):
Multiply circumference by height
circumference
height
Take a can of soup
Peel off the label
To find total area (T.A.): Take the label (LA)
Pop the top and find the area of the base
Top Bottom
Add 2 base areas to the LA
A = πr²
To find volume (V): Start with the area of the base
Multiply it by height
H
r
That’s how much soup is in the can !
A = πr²
Lateral Area of a Cylinder: L.A.
The lateral area of a cylinder equals the circumference of a base times the height of the cylinder.
L.A = 2πrH
LA = 12π • 8 = 96π square units
L.A = dπH
L.A = CH
which is
which is
68
•
Total Area of a Cylinder: T.A.
The total area of a cylinder is the lateral area plus twice the area of a base.
T.A = L.A. + 2B
TA = 96π + 2(π • 6²) = 96π + 2(36π) = 96π + 72π = 168π square units
68
•
T.A. = 2πrH + 2πr²
which is
Volume of a Cylinder: V
The volume of a cylinder equals the area of a base times the height of the cylinder.
V = πr²H
V = (π • 6²) • 8 = 36π • 8 = 288π cubic units
68
•
Lateral Area of a Cone: L.A.
LA = π • 6 •10 = 60π square units
L.A = πrl 6
10
•
8
Total Area of a Cone: T.A.
T.A = L.A. + B
TA = 60π + (π • 6²) = 60π + 36π = 96π square units
T.A. = πrl + πr²
which is
6
10
•
8
Volume of a Cone: V
The volume of a cone equals one third the area of the base times the height of the cone.
V = πr²h
V = 1/3 • π • 6² • 8 = 96π cubic units
6
10
•
81
3
12.4 Spheres
Surface Area Formula24 rSurface Area =
r
What does this represent?It represents the area of a circle with radius = r.
Volume Formula
Volume =34
3r
A way to remember which formula is cubed…The volume formula has two 3’s and volume is measured in units3!!
1. radius = 10 Area: _____ Volume: _____
24 rSurface Area =
= 4π(10)2
= 4π(100)
= 400π units2
Volume =34
3r
3)10(3
4
10003
4
3units 3
4000
Geometry
12.5 Areas and Volumes of Similar Solids
If the scale factor of two solids is a:b, then
(1) the ratio of corresponding perimeters is a:b
(2) the ratio of base areas, of lateral areas, and of the total area is a²:b²
(3) the ratio of volumes is a³:b³
Scale Factor
SCALE FACTOR: 1:2Base circumference: 6π:12π 1:2Lateral areas: 15π:60π 1:4Volumes: 12π:96π 1:8
6
10
•
8
3
4 5
Exercises
Find the missing information.
4. 5. 6. 7. 8.
scale factor 2 : 5
ratio of base perimeters
ratio of heights 1 : 3
ratio of lateral areas 4 : 49
ratio of total areas
ratio of volumes 125 : 216
27 : 1000
2:5
2:5
4:25
4:25
8:125
1:9
1:27
2:7
8:343
5:6
25:36
3:10
9:100
HW
• Midpoint WS
