geometry surface area of pyramids and cones. december 8, 2015 goals know what a pyramid is. find the...
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Geometry
Surface Area of Pyramids and Cones
April 21, 2023
Goals
Know what a pyramid is. Find the surface area of a pyramid. Know what a cone is. Find the surface area of a cone.
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Pyramids
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Pyramid
Polyhedron with a polygon base. Lateral faces are triangles. Lateral faces meet at a common vertex. Regular pyramid: base is a regular
polygon. (This is what we will discuss.)
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Square PyramidVertex
Altitude (h) Slant Height (l)
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Surface Area of a Pyramid
The surface area is the sum of the area of the base and the area of the lateral faces.
SA = B + LA
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Surface Area of a Pyramid
Unfold the pyramid:
The surface area is the sum of the area of the base and the area of the lateral faces.
S
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Lateral Area
Re-arrange the triangles.
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Lateral Area
Find the area of a triangle.
S
?S S S S
?
12A sl
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Lateral Area
Find the area of four triangles.
S S S S
12
12
4
(4 )
A sl
s l
What is 4s?
Perimeter!
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Lateral Area of a Pyramid
12LA Pl
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Surface Area of a Pyramid
S
12
12
LA Pl
SA B Pl
Just add the area of the base to the LA.
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Example Find the LA & the SA of the square pyramid.
12
16
B = 122 = 144
P = 4(12) = 48
l = 16
12
12144 48 16
144 384
528
SA B Pl
LA
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Example 2 Find the surface area of the square pyramid.
8
6
B = 82 = 64
P = 4(8) = 32
l = ?
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Example 2 Finding the slant height.
8
6
?4
l
2 2 24 6
16 36
52
52 2 13
7.21
l
l
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Example 2 Finding the SA
8
6 7.21
12
2 128 32 7.21
179.38
64 115.38
SA B Pl
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Your Turn Find the SA of the square pyramid.
10
12
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Finding the slant height.
10
12
5
2 2 25 12
25 144
169
1169 3
l
l
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Finding the surface area.
10
12
5
13
12
2 1210 40 13
100 260
360
SA B PL
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Example Find the lateral area.
20
18
10
2 2 2
1
21
802
10 18
14.96
180 14.96 598.7
2
A Pl
A l
l
l
A
14.96
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Cones
Circular base No Lateral Faces SA = B + LA Can be right or
oblique
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Cone Surface Area
Cut the cone from the vertex to the base.
Then around the base.
You get:
Circular Base
Circle Sector
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Cone Surface AreaA = r2
Circular Base
Circle Sector
r
l
lThe length of this arc is the circumference of the base.
2r
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Sector Area
Circle Sector
l
2r
2
2
2
2
2
2
A r
l l
r lA
lA rl
This is the lateral area of a cone.
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Cone Surface AreaA = r2
Circular Base
Circle Sector
r
l
l
2r A = rl
SA = r2 + rl
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Cone Surface Area
r
lSA = r2 + rl
SA = r(r + l)
LA = rl
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Example Find the SA of the cone.
4
6SA = r(r + )
SA = (4)(4 + 6)
SA = 40
SA 125.66
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Your Turn Find the SA of the cone.
5
15
SA = (5)(5 + 15)
SA = (5)(20)
SA = 100
SA 314.16
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Example Find the SA of the cone.
8
3
l
Finding l:2 2 28 3
64 9
73
73 8.54
l
l
8.54
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Example Find the SA of the cone.
8
3
Finding SA:
SA = r(r + )
SA = (3)(3 + 8.54)
SA = (3)(11.54)
SA 108.76
8.54
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Your Turn Find the surface area
12
30
L2 = 302 + 122
L 32.31
SA = π(12)(12 + 32.31)
SA = 1670.45
32.31
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Now you can find the surface area and lateral area of cone or a pyramid!
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Summary
12
12
2
SA B Pl
LA Pl
SA r rl
r r l
LA rl
r
l
Sl
These should be on your formula sheet.
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Practice Problems