lecture 6 - pyramid rev
TRANSCRIPT
8/19/2019 Lecture 6 - Pyramid Rev
http://slidepdf.com/reader/full/lecture-6-pyramid-rev 1/19
MALAYAN COLLEGES
LAGUNA
8/19/2019 Lecture 6 - Pyramid Rev
http://slidepdf.com/reader/full/lecture-6-pyramid-rev 2/19
MALAYAN COLLEGES LAGUNA
MATH 014
SOLID MENSURATION
2
8/19/2019 Lecture 6 - Pyramid Rev
http://slidepdf.com/reader/full/lecture-6-pyramid-rev 3/19
MALAYAN COLLEGES LAGUNA
PYRAMID
pyramid is a polyhedron of which one fac
e, called the base, is a polygon of any numb
er of sides and the other faces are triangles
which have a common vertex.
3
8/19/2019 Lecture 6 - Pyramid Rev
http://slidepdf.com/reader/full/lecture-6-pyramid-rev 4/19
MALAYAN COLLEGES LAGUNA
PYRAMID
pyramid is a polyhedron of which one fac
e, called the base, is a polygon of any numb
er of sides and the other faces are triangles
which have a common vertex.
4
Properties
1. The triangular facesare called lateral faces.
2. The common vertexis also called the apex.
8/19/2019 Lecture 6 - Pyramid Rev
http://slidepdf.com/reader/full/lecture-6-pyramid-rev 5/19
MALAYAN COLLEGES LAGUNA
PYRAMID
2. The altitude (or height) of a pyramid is t
he length of the perpendicular dropped fro
m the vertex to the plane of then base.
3. The slant height or slant
length of a triangular face
is the height of that
triangle.
5
8/19/2019 Lecture 6 - Pyramid Rev
http://slidepdf.com/reader/full/lecture-6-pyramid-rev 6/19
MALAYAN COLLEGES LAGUNA
PYRAMID
ight vs !bli"ue #yramid
This tells you where the apex is. $f the apex
is directly above the center of the base, the
n it is a ight #yramid, otherwise it is an !b
li"ue #yramid.
6
8/19/2019 Lecture 6 - Pyramid Rev
http://slidepdf.com/reader/full/lecture-6-pyramid-rev 7/19
MALAYAN COLLEGES LAGUNA
PYRAMID
egular vs $rregular #yr
amid
This tells us about the
shape of the base.
$f the base is a regular
polygon, then it is a e
gular #yramid, otherwi
se it is an
$rregular #yramid.
7
8/19/2019 Lecture 6 - Pyramid Rev
http://slidepdf.com/reader/full/lecture-6-pyramid-rev 8/19
MALAYAN COLLEGES LAGUNA
PYRAMID
3. $f a pyramid is cut by
a plane parallel to the b
ase, the lateral edges (a
nd bases) and the altitu
de are divided proporti
onally%
$n symbols we write&
' * x
h
8
8/19/2019 Lecture 6 - Pyramid Rev
http://slidepdf.com/reader/full/lecture-6-pyramid-rev 9/19
8/19/2019 Lecture 6 - Pyramid Rev
http://slidepdf.com/reader/full/lecture-6-pyramid-rev 10/19
MALAYAN COLLEGES LAGUNA
PYRAMID
ight egular #yramid is one whose base is a regular polygon wh
ose center coincides with the foot of the perpendicular dropped f
rom the vertex to the base.
#roperties&
. The lateral edges of a regular pyramid
are e"ual.
2. The lateral faces of a regular pyramid
arecongruent isosceles triangles.
3. The altitudes of the lateral faces of a
regular pyramid are e"ual.
. The slant height of a regular pyramid
is the altitude of the lateral face.
4. The altitude of a regular pyramid is e"ual
to the length of the perpendicular dropped
from the vertex to the center of the base.
10
8/19/2019 Lecture 6 - Pyramid Rev
http://slidepdf.com/reader/full/lecture-6-pyramid-rev 11/19
MALAYAN COLLEGES LAGUNA
PYRAMID
+!- /&
1olume * 3 b ase x alt itude
The lateral area of a right regu
lar pyramid is e"ual to one0hal
f the product of the perimeter
of the base and the slant heig
ht.
ateral rea * 5 per imeter of
base x s lant height
11
8/19/2019 Lecture 6 - Pyramid Rev
http://slidepdf.com/reader/full/lecture-6-pyramid-rev 12/19
MALAYAN COLLEGES LAGUNA
PYRAMID
+ind the area of the base of a regular s"uare pyramid whose la
teral faces are e"uilateral triangles and whose altitude of the p
yramid is 6 in. (s*6s"rt2% 26s". in. )
The base of a regular pyramid is a regular hexagon, each of wh
ose side is 6 cm. $f the slant height of the pyramid is 7 cm., w
hat is its a) lateral area% b) total surface area% c) volume8
( ns & 27 s". cm.% 79.32 s". cm.% 3::.4 cu.cm )
pyramid whose base is an e"uilateral triangle has a volume o
f 67;3 cu. cm. $f the altitude of the pyramid is 7 cm., what is
the length of the sides of the base8 ( ns & : .67 cm. )
12
PROBLEM!
The monument of "estius in Rome#$hich is a s%uare p&ramid 121 ' ft.high $ith a (ase edge measuring )*.+ft. ,ind the num(er of s%uare feet inthe lateral surface of the monument.-hat is its volume ( Ans: 25,797.23sq. ft. / 392,143.68 cu. ft )
8/19/2019 Lecture 6 - Pyramid Rev
http://slidepdf.com/reader/full/lecture-6-pyramid-rev 13/19
MALAYAN COLLEGES LAGUNA
PYRAMID
+ind the area of the base of a regular s"uare pyr
amid whose lateral faces are e"uilateral triangles
and whose altitude of the pyramid is 6 in. (s*6s"rt
2 % 26s". in . )
13
e
e/2 e/2
es
8i
8/19/2019 Lecture 6 - Pyramid Rev
http://slidepdf.com/reader/full/lecture-6-pyramid-rev 14/19
MALAYAN COLLEGES LAGUNA
PYRAMID
#!</&
$f there are = cu. ft. in a bushel, what
is the capacity ( in bushels ) of a hopper i
n the shape of an inverted pyramid 2 f
t. deep and 6 ft. s"uare at the top8
( ns& 2
7.6 bushels )
church spire in the form of a regular h
exagonal pyramid whose base edge is 6 f
t. and whose altitude is >4 ft. is to be pai
nted at a cost of #h# 22.77 per s"uare ya
rd. ?hat is the total cost8 ( ns& #h#
:.77 )
14
8/19/2019 Lecture 6 - Pyramid Rev
http://slidepdf.com/reader/full/lecture-6-pyramid-rev 15/19
MALAYAN COLLEGES LAGUNA
PYRAMID church spir
e in the form
of a regular h
exagonal pyra
mid whose ba
se edge is 6 f
t. and whose
altitude is >4
ft. is to be pai
nted at a cost
of #h# 22.77
per s"uare ya
rd. ?hat is th
e total cost8
( ns& #h#
:.77 )
15
8/19/2019 Lecture 6 - Pyramid Rev
http://slidepdf.com/reader/full/lecture-6-pyramid-rev 16/19
MALAYAN COLLEGES LAGUNA
PYRAMID
#!<&The regular pyramidal roof of the ?ashington
onument is 44 ft. high and has a base which is a
s"uare 34 ft. 2 in. on a side. The marble slabs of
which it is built weigh 94 lbs. per cu. ft. $f the r
oom covered by the roof is a pyramid whose s"u
are base is 3 ft. on a side and 4 ft. high, what i
s the weight of the roof8 ( ns& 37>,977 lbs. )
16
hidden
8/19/2019 Lecture 6 - Pyramid Rev
http://slidepdf.com/reader/full/lecture-6-pyramid-rev 17/19
MALAYAN COLLEGES LAGUNA
PYRAMID
#!</&
The altitude of the great #yrami
d of @heops in <gypt originally w
as 67 ft. and its s"uare base wa
s >9 ft. on an edge. $t is said to
have cost / 7 a cubic yard and
/3 more for each s"uare yard of
lateral surface. ?hat was its cos
t8
( ns& / 3,:7,777 )
17
The roof of a $ater to$er is composed of / e%ualisosceles triangles $hose vertices meet in the center ofthe roof. 0f the inclined edges measures 1 ft. and theheight of the roof is * ft.# nd the num(er of s%uare feetof the tar paper necessar& to cover the roof. 3eglect the$aste in lapping# cutting# etc. ( Ans: 686.53 sq. ft. )
8/19/2019 Lecture 6 - Pyramid Rev
http://slidepdf.com/reader/full/lecture-6-pyramid-rev 18/19
MALAYAN COLLEGES LAGUNA
PYRAMID
#!<&
+ind the volume of the largest pyramid which c
an be cut from a rectangular prism whose edge
s are 2 in. by 3 in. by in. Aiscuss fully.
( ns& 6 cu. in .)
18
8/19/2019 Lecture 6 - Pyramid Rev
http://slidepdf.com/reader/full/lecture-6-pyramid-rev 19/19
MALAYAN COLLEGES LAGUNA19
e!"#