10­5 Surface Area of Pyramids and Cones
A. parts of a pyramid
1. The vertex of the pyramid is a point opposite the base of the pyramid.
2. The base of a regular pyramid is a regular polygon.
3. The lateral faces are congruent isosceles triangles.
4. The slant height of a regular pyramid is the distance from the vertex to the midpoint of an
edge of the base.
5. The altitude of a pyramid is a perpendicular segment from the vertex to the plane of the
base.
Vertices
lateral faces
altitude
bases
Regular Pyramid
nonregular pyramid
B. Formulas for lateral and surface area of regular
pyramids.
1. Deriving the Formulas for Lateral and Surface Area
2. Lateral and Surface Area of a Regular Pyramid
a. Lateral area of a regular pyramid with perimeter P and slant height l is L=½Pl.
b. Surface area of a regular pyramid with lateral
area L and base area B is s = L + B or S= ½Pl
+ B.
3. Example 1­­finding lateral area and surface area of
pyramids
Find the lateral area and surface area of each regular pyramid. Round to the nearest 10th, if necessary.
a. A regular square pyramid with base edge length 14 cm and slant height 25 cm
L = _______ S = _______
b. L = _______ S = _______
C. Parts of a Cone
1. The vertex of a cone is a point opposite the base.
2. The axis of a cone is a segment with endpoints at the vertex and the center the base.
3. The axis of a right cone is perpendicular to the base.
4. The axis of an oblique cone is not perpendicular to the base.
Vertices
lateral surfaces
axis
axis
bases
Right cone
oblique cone
5. The slant height of a regular cone is the distance from the vertex of a right cone to a point on the edge of the base.
6. The altitude of a cone is a perpendicular segment from the vertex of the cone to the plane of the base.
D. Formulas for lateral and surface area of right cones
1. The lateral area of a right cone with radius r and slant height l is L = πrl
2. The surface area of a right cone with lateral area L and base area B is S = L + B or
S = πrl + πr2.
3. Example 2­­finding lateral area and surface area of
right cones
Find the lateral area and surface area of each cone.
a. A right cone with radius 9 cm and slant height 5 cm
b. 4. Example 3­­exploring effects of changing dimensions
The base edge length and slant height of the regular hexagonal pyramid are both divided by
5. Describe the effect on the surface area.
5. Example 4­­finding surface area of composite 3­D figures
Find the surface area of the composite figure.