Download Surface area of a cone

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TEKS: G2B, G3B, G6B, G8D, G11D
The student will make conjectures about 3-D
figures and determine the validity using a variety
of approaches.
The student will construct and justify statements
about geometric figures and their properties.
The student will use nets to represent and
construct 3-D figures.
The student will find surface area and volume of
prisms, cylinders, cones, pyramids, spheres, and
composite figures.
The student will describe the effect on perimeter,
area, and volume when one or more dimensions
of a figure are changed.
The vertex of a cone is the point opposite the
base. The axis of a cone is the segment with
endpoints at the vertex and the center of the
base. The axis of a right cone is perpendicular
to the base. The axis of an oblique cone is not
perpendicular to the base.
The slant height of a right cone is the distance from the
vertex of a right cone to a point on the edge of the base.
The altitude of a cone is a perpendicular
segment from the vertex of the cone to the plane of the
base.
Example: 1
Find the lateral area and surface area of a right cone
with radius 9 cm and slant height 5 cm.
L = rℓ
= (9)(5)
Lateral area of a cone
Substitute 9 for r and 5 for ℓ.
= 45 cm2
S = rℓ + r2
= 45 + (9)2
= 126 cm2
Surface area of a cone
Substitute 5 for ℓ and 9 for r.
Example: 2
Find the lateral area and surface area of the cone.
Use the Pythagorean Theorem to find ℓ.
L = rℓ
= (8)(17)
= 136 in2
S = rℓ + r2
= 136 + (8)2
= 200 in2
Lateral area of a right cone
Substitute 8 for r and 17
for ℓ.
Surface area of a cone
Substitute 8 for r and 17
for ℓ.
Example: 3
Find the surface area of the
composite figure.
Left-hand cone:
The lateral area of the cone is
L = rl = (6)(12) = 72 in2.
Right-hand cone:
Using the Pythagorean Theorem, l = 10 in.
The lateral area of the cone is
L = rl = (6)(10) = 60 in2.
Composite figure:
S = (left cone lateral area) + (right cone lateral area)
= 60 in2 + 72 in2 = 132 in2
Example: 4
Find the lateral area and surface area of the right cone.
Use the Pythagorean Theorem to find ℓ.
ℓ
L = rℓ
= (8)(10)
= 80 cm2
S = rℓ + r2
= 80 + (8)2
= 144 cm2
Lateral area of a right cone
Substitute 8 for r and 10
for ℓ.
Surface area of a cone
Substitute 8 for r and 10
for ℓ.
Example: 5
If the pattern shown is used to
make a paper cup, what is the
diameter of the cup?
The radius of the large circle used to create the
pattern is the slant height of the cone.
The area of the pattern is the lateral area of the cone. The area of the
pattern is also
of the area of the large circle, so
Example: 5 cont.
If the pattern shown is used to make a paper
cup, what is the diameter of the cup?
Substitute 4 for ℓ, the slant
height of the cone and the
radius of the large circle.
r = 2 in.
Solve for r.
The diameter of the cone is 2(2) = 4 in.
Example: 6
What if…? If the radius of the large circle were 12 in.,
what would be the radius of the cone?
The radius of the large circle used to create the pattern is the slant
height of the cone.
The area of the pattern is the lateral area of the
cone. The area of the pattern is also
of the area of
the large circle, so
Substitute 12 for ℓ, the slant height of the
cone and the radius of the large circle.
r = 9 in.
Solve for r.
The radius of the cone is 9 in.