Transcript Surface Area of Pyramids and Cones

Drill
1) Find the height of a rectangular prism
with a given length of 6 feet a width of 5
feet and a volume of 330 cubic feet?
2) What is the lateral area of a triangular
prism where the three sides of the base
are 6, 8, and 10 respectively and the
height of the prism is 12 feet?
3) What is the surface area of the same
triangular prism if the base is a right
triangle and 8 & 6 are the legs of the
base?
Objectives
• Find the surface area of a pyramid.
• Find the surface area of a cone.
6.3
Surface Area
of Pyramids and Cones
Finding the surface area of a
pyramid
• A pyramid is a polyhedron in which the base is a
polygon and the lateral faces are triangles with a
common vertex. The intersection of two lateral
faces is a lateral edge. The intersection of the
base and a lateral face is a base edge. The
altitude or height of a pyramid is the
perpendicular distance between the base and
the vertex.
Vocabulary
Pyramid: a pyramid consists of one
base and then triangles for lateral
faces.
Altitude: is the length of the segment
perpendicular from the vertex to the
base.
Slant Height: the slant height of a
pyramid is the height of one lateral
face.
More on pyramids
• A regular pyramid has a
regular polygon for a
base and its height meets
the base at its center.
The slant height of a
regular pyramid is the
altitude of any lateral
face. A nonregular
pyramid does not have a
slant height.
Pyramid Arena
Lateral Area of a Right Regular
Pyramid
The lateral area of a pyramid is the
sum of all the areas in the lateral
faces.
L = ½ lp
* Where “l” is the slant height and “p” is
the perimeter of the base.
Vocabulary
Surface Area: The surface area
“S” of a pyramid with lateral area
“L” and area of a base “B” is:
S=L+B
Surface Area of a Pyramid
Example: The roof of a gazebo is a
square pyramid, if one side of the
square base is 12 feet and the slant
height is 16 feet. Find the lateral area
of the roof.
* If the materials cost $3.50 per sq. ft.
how much will it cost to build the roof?
Ex. 1: Finding the Area of a Lateral
Face
• Architecture. The lateral faces of the
Pyramid Arena in Memphis, Tennessee,
are covered with steal panels. Use the
diagram of the arena to find the area of
each lateral face of this regular pyramid.
Hexagonal Pyramids
• A regular hexagonal
pyramid and its net are
shown at the right. Let b
represent the length of a
base edge, and let l
represent the slant height
of the pyramid. The area
of each lateral face is
1/2bl and the perimeter
of the base if P = 6b. So
the surface area is as
follows:
Hexagonal pyramid
S = (Area of base) + 6(Area of lateral face)
S = B + 6( ½ bl)
S = B + (6b)l
Substitute
Rewrite 6( ½ bl) as ½ (6b)l.
S = B + Pl
Substitute P for 6b
Surface Area of a Regular Pyramid
The surface area S of a regular pyramid is:
S = B + ½ Pl, where B is the area of the base, P is
the perimeter of the base, and l is the slant height.
Ex. 2: Finding the surface area of
a pyramid
• To find the surface area
of the regular pyramid
shown, start by finding
the area of the base.
• Use the formula for the
area of a regular
polygon,
½ (apothem)(perimeter).
A diagram of the base
is shown to the right.
Ex. 2: Finding the surface area of
a pyramid
After substituting, the
area of the base is
3
½ (3
)(6• 6), or
543
square meters.
Surface area
• Now you can find the surface area by
using 54 3for the area of the base, B.
Vocabulary
Cone: is an object that consists of
a circular base and a curved
lateral surface which extends
from the base to a single point
called the vertex.
Finding the Surface Area of a Cone
• A circular cone, or cone, has
a circular base and a vertex
that is NOT in the same
plane as the base. The
altitude, or height, is the
perpendicular distance
between the vertex and the
base. In a right cone, the
height meets the base at its
center and the slant height
is the distance between the
vertex and a point on the
base edge.
Finding the Surface Area of a Cone
• The lateral surface of a
cone consists of all
segments that connect the
vertex with points on the
base edge. When you cut
along the slant height and
like the cone flat, you get
the net shown at the right.
In the net, the circular base
has an area of r2 and the
lateral surface area is the
sector of a circle.
More on cones . . .
• You can find the area of this sector by
using a proportion, as shown below.
Area of sector
Arc length
Area of circle = Circumference Set up proportion
2r
Area of sector
Substitute
=
2
l
2l
2r Multiply each side by l2
Area of sector = l2 • 2l
Area of sector = rl
Simplify
The surface area of a cone is the sum of the base
area and the lateral area, rl.
Lateral Area of a Cone
The lateral area of a cone is equal to:
L  rl
* Where “r” is the radius of the base and
“l’ is the slant height of the cone.
Surface Area of a Cone
The surface area of a cone is equal
to:
S= L + B
S  rl  r
2
* Where “r” is the radius of the base
and “l’ is the slant height of the cone.
Ex. 3: Finding the surface area of
a cone
• To find the surface area
of the right cone shown,
use the formula for the
surface area.
S = r2 + rl
Write formula
S = 42 + (4)(6)
Substitute
S = 16 + 24
Simplify
S = 40
Simplify
The surface area is 40 square inches or about
125.7 square inches.