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Section 8.2 Pyramids, Area, & VolumePowerPoint Presentation

Section 8.2 Pyramids, Area, & Volume

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Section 8.2 Pyramids, Area, & Volume

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Section 8.2Pyramids, Area, & Volume

Section 8.2 Nack/Jones

- 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

- 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.

l

h

a

Section 8.2 Nack/Jones

- 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

- 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 = 42

Find Lateral Area:

L = ½Pl= ½42 (16) = 32 2

Find the area of the Base: 6

B = 16 l

Total Area = 16 + 32 22

6

4

Section 8.2 Nack/Jones

- 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,

we know that a = 23

B = ½ 23 (64) = 24 3

V =1/3 Bh = 96 3 units3

=12

30

60

4

Section 8.2 Nack/Jones