lecture 6 - pyramid rev

8/19/2019 Lecture 6 - Pyramid Rev

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MALAYAN COLLEGES

LAGUNA



MALAYAN COLLEGES LAGUNA

MATH 014

SOLID MENSURATION

2




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




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.




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




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.

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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.

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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

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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.

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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

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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. )

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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 )




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




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




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




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. )

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hidden




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. )




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 .)

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MALAYAN COLLEGES LAGUNA19

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lecture 6 - pyramid rev

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