12-5 Surface Area of Pyramids

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12-5 Surface Area of Pyramids
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Objectives

• Find lateral areas of regular pyramids
• Find surface areas of regular pyramids

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Characteristics

• All faces except the base intersect at one point
  called the vertex
• The base is always a polygon
• The faces that intersect at the vertex are called
  lateral faces and form triangles
• The edges of the lateral faces that have the
  vertex as an endpoint are all lateral edges
• The altitude is the segment from the vertex
  perpendicular to the base

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Parts of a Pyramid
Vertex
Lateral Edge
Lateral Face
Base
Altitude
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FYI
Lateral area of Regular pyramids can be found by
adding the area of all its congruent triangular
faces
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Formula for Lateral Area of a Pyramid

• If a regular pyramid has a lateral area of L
  square units, a slant height of l units, and its
  base has a perimeter of P units
• then
• L1/2(P)(l )

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Example 1
A regular octagonal pyramid has a side of 25
kilometers and a slant height of 75 kilometers.
Find the lateral area of this figure.
75 Km
25 Km
Top-Down view
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The Work
L1/2(P)(l)
P200
Slant height (l)75
L1/2(200)(75)
75 Km
L1/2(15000)
25 Km
Lateral Area7500 km²
Top-Down view
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Example 1 Explained

• Find the slant height
• Side of base 25 so the perimeter (P) is 200 km
• Slant height 75 km
• Formula for Lateral Area is L1/2 (P)(l)
• P200 and l75
• L1/2(200)(75)
• L1/2(15000)
• L7500km²

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Formula For Surface Area of a Regular Pyramid

• If a regular pyramid has a surface area of T
  square units, a slant height of l units and its
  base has a perimeter of P units and area of B
  square units,
• then
• T1/2(P)(l ) B

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• The slant height of the pyramid is the hypotenuse
  of a right triangle with legs that are the
  altitude from the vertex and a segment with a
  length that is one half of the side measure of
  the base.
• Because of this you can use the Pythagorean
  Theorem to find a missing side.

Slant Height (the hypotenuse)
Altitude (a side of the
triangle)
Segment (like a radius of a circle because it is
half the length of the side of a square-also a
side of the triangle)
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Example 2
A square pyramid has an altitude of 72 fathoms
and a length of one side of the base being 54
fathoms. Before finding the surface area find the
slant height.
l
72 Fathoms
54 Fathoms
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The Work
L1/2(P)(l)B
27² 72² c²
729 5184 c²
Perimeter54x4 or 216 fa
v5913 vc²
Area54x54 or 2916 fa²
c76.89 so l76.89 fa
T1/2(216)(76.89)2916
l
T1/2(16608.24)2916
72 Fathoms
T8304.122916
T11220.2 Fathoms²
54 Fathoms
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Example 2 Explained

• First Find the Slant Height
• The segment from the center of the pyramid
• to the side is like a radius so it 27 fm
• Use Pythagorean theorem 27² 72² c²
• Cv3913 or 76.9 fathoms which is the slant height
  (l)
• Second Find the Surface Area
• Find the Area and perimeter of base
• Area (B)2916 Perimeter (P) 216
• Use the Formula T1/2(P)(l)B
• T1/2(216)(76.9)2916
• T1/2(16610.4)2916
• T8305.22916
• T11221.2 fathoms²

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example 3-Surface Area of a Pentagonal Pyramid
13 ft
20 ft
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1st-find the missing segment2nd-find ½ the
length of one side of the base using
Trigonometry3rd-find the area and perimeter of
the base4th-use this to find the surface area
a²13²20²
A15.2 ft
13 ft
Tan(36)x/15.2
X11 ft
20 ft
Perimeter5(211)
Area1/2(22)(15.2)5
Length of segment from Pyramid (A)
Area836
P110 ft
T1/2(P)(l)B
T1/2(110)(20)836
36º (Half of the central angle)
T(Surface Area)1936 ft²
11 ft
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Example 3 Explained

• First Use Pythagorean Theorem to find the missing
  side
• Pythagorean Theorem 13² b² 20²
• 169 b² 400
• b²231
• b15.2 ft
• Now Find the Length of the sides of the base
  using Trig
• The central angle of a pentagon is 72º so half of
  that is 36º
• The side of the triangle is the missing segment
  you found earlier (b)
• Using this information you can set up a trig
  equation
• Tan 36º x/15.2
• 15.2 (tan 36º)x
• X 11 ft
• Next find the perimeter and area of the base
• P5(112)
• P110 ft
• A1/2(15.2)(22)5
• A836
• Finally find the surface area
• The Formula for S.A. is T1/2PlB

15.2 ft
36º (Half of the central angle)
x
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Assignment
Pg 663-667 7-39 evens 42-45

• Pg. 663
• 7-15, 18-23
• Omit 20