Transcript Surface Areas of Cones

Surface Areas of Cones
– Find the lateral areas of cones.
– Find surface areas of cones.
A cone-shaped teepee in a
Shoshone camp, South Pass,
Wyoming 1870
LATERAL AREAS OF CONES
Cones have the following
characteristics:
• The base is a circle and the vertex
is the point V.
• The axis is the segment whose
endpoints are the vertex and the
center of the base.
• The altitude is the segment from
the vertex perpendicular to the
base of the cone.
LATERAL AREAS OF CONES
The axis is also
an altitude
Axis
Oblique Cone
Altitude
Slant
height
l
Right Cone
LATERAL AREAS OF CONES
We can use the net for the cone to derive the formula for the
lateral area.
r
l
r
l
LATERAL AREAS OF CONES
•The lateral region of the cone is a sector of a
circle with radius l
•The arc length of the sector is the same as
the circumference of the base, or 2r.
•The circumference of the circle containing
the sector is 2l
l
r
•The area of the sector is proportional to the
area of the circle.
r
l
LATERAL AREAS OF CONES
•The area of the sector is proportional to the
area of the circle.
l
area of sector =
measure of arc
area of circle = circumference of circle
r
area of sector = 2r
2l2
= 2l
(l
area of sector =
2l
r
2)(2r)
area of sector = rl
l
Key Concept
Lateral Area of a Cone
If a right circular cone has a lateral of L
square units, a slant height of l units, and
the radius of the base is r units, then
L = rl
l
r
Example 1
Lateral Area of a Cone
LAMPS Diego has a conical lamp shade with an altitude of 6
inches and a diameter of 12 inches. Find the lateral area of
the lamp shade.
6 in.
12 in.
SURFACE AREAS OF CONES
To find the surface area of a cone, add the area of the base
to the lateral area.
Key Concept
Surface Area of a Cone
If a right circular cone has a surface area of T square units,
a slant height of l units, and the radius of the base is r units,
then
T = rl + r2
l
r
Example 2
Surface Area of a Cone
13.6 cm
4.7 cm.