8.2

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Transcript 8.2 

Section 8.2
Pyramids, Area, & Volume
Section 8.2 Nack/Jones
1
Pyramid
• The solid figure formed by connecting a polygon with a
point not in the plane of the polygon is called a pyramid.
• The polygonal region is called the base & the point is the
vertex.
• A regular pyramid is a pyramid whose base is a regular
polygon and whose lateral edges are all congruent.
• The slant height of a regular pyramid is the altitude from
the vertex of the pyramid to the base of any of the
congruent lateral faces of the regular pyramid.
• The line segment from the vertex perpendicular to the
plane of the base is the altitude.
Section 8.2 Nack/Jones
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Pyramid
• In the regular pyramid, the
distance l is called the slant
height of the lateral surfaces of a
regular pyramid.
• Theorem 8.2.1: In a regular
pyramid, the length a of the
apothem of the base, the altitude
h, and the slant height l satisfy
the Pythagorean Theorem, that is
l² = a² + h² in every regular
pyramid.
Section 8.2 Nack/Jones
l
h
a
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LateralSurface Area of a Pyramid
• Theorem 8.2.2: The
Lateral Area L of a regular
pyramid with slant height l
and perimeter P of the
base is given by:
L = ½ pl
It is simpler to find the area
of one lateral face and
multiply by the number of
faces.
Example 2 p. 401
Section 8.2 Nack/Jones
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Total Surface Area
• Theorem 8.2.3: The total area (surface area) T of a pyramid with
lateral area L and base area B is given by ( the sum of the area of all
its faces):
T = L + B or T = ½ Pl + B
Example:
To find the total area,
Find the slant height. Apply Pythagorean Theorem to one face:
l ² + 2² = 6² or l = 42
Find Lateral Area:
L = ½ Pl = ½ 42 (16) = 32 2
Find the area of the Base:
B = 16
Total Area = 16 + 32 2
6
6
l
2
Section 8.2 Nack/Jones
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5
Volume of a pyramid
• Theorem 8.2.4: The volume V
of a pyramid having a base
area B and an altitude of length
H is given by:
V =1/3 Bh
Example:
Find the area of the base:
B = ½aP.
Since it is a 30-60-90 triangle, 60
4
we know that a = 23
B = ½ 23 (64) = 24 3
V =1/3 Bh = 96 3 units3
Section 8.2 Nack/Jones
=12
30
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