Download Notes: Surface Area of Pyramids and Cones

Notes: Surface Area of Pyramids
and Cones
I. Pyramids
The vertex of a pyramid is the point opposite the base of the
pyramid. The base of a regular pyramid is a regular polygon, and the
lateral faces are congruent isosceles triangles. The slant height of a
regular pyramid is the distance from the vertex to the midpoint of
an edge of the base. The altitude of a pyramid is the perpendicular
segment from the vertex to the plane of the base.
The lateral faces of a regular pyramid can be arranged to cover
half of a rectangle with a height equal to the slant height of the
pyramid. The width of the rectangle is equal to the base
perimeter of the pyramid.
Ex 1a. Find the lateral area and surface area of a
regular square pyramid with base edge length 14
cm and slant height 25 cm. Round to the nearest
tenth, if necessary.
Lateral area of a regular pyramid
P = 4(14) = 56 cm
Surface area of a regular pyramid
B = 142 = 196 cm2
Ex 1b: Find the lateral area and surface area of a regular triangular
pyramid with base edge length 6 ft and slant height 10 ft. The base
area is 𝟗 𝟑 ft2.
Step 1 Find the base perimeter.
Step 2 Find the lateral area.
Step 3 Find the surface area.
The base perimeter is 3(6) = 18 ft.
II. Cones
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.
Ex 2a: Find the lateral area and surface area of
a right cone with radius 9 cm and slant height 5
cm.
L = rℓ
Lateral area of a cone
= (9)(5) = 45 cm2
S = rℓ + r2
Substitute 9 for r and 5 for ℓ.
Surface area of a cone
= 45 + (9)2 = 126 cm2
Substitute 5 for ℓ and 9
for r.
Ex 2b: Find the lateral area and surface area of the cone.
l = 17
L = rℓ
= (8)(17)
= 136 in2
Lateral area of a right cone
Substitute 8 for r
and 17 for ℓ.
S = rℓ + r2
= 136 + (8)2
= 200 in2
Surface area of a cone
Substitute 8 for r
and 17 for ℓ.
Ex 2c: Find the lateral area and surface area of the right cone. The
slant height is 10cm.
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.