Honors Geometry
Lesson 12.3
Surface Area of Pyramids and
Cones
What You Should Learn
Why You Should Learn It
 How to find the surface area of a
pyramid
 How to find the surface area of a cone
 Many real-life objects are pyramids or
cones, such as cone-shaped squirrel
barrier for a bird feeder
Pyramids
 A pyramid is a polyhedron in which the
base is a polygon and the lateral faces
are triangles that have a common
vertex
Pyramids
 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
the pyramid is the
perpendicular distance
between the base and
the vertex
Regular Pyramid
 A pyramid is regular if its base is a regular
polygon and if the segment from the vertex to
the center of the base is perpendicular to the
base
 The slant height of a regular pyramid is the
altitude of any lateral face (a nonregular pyramid has
no slant height)
Developing the formula for surface
area of a regular pyramid
 Area of each triangle is ½bL
 Perimeter of the base is 6b
 Surface Area =
(Area of base) + 6(Area of lateral faces)
 S = B + 6(½bL)
 S = B + ½(6b)(L)
 S= B + ½PL
Theorem 12.4
Surface Area of a Regular Pyramid
 The surface area, S, of a regular
pyramid is
S = B + ½PL
Where B is the area of the base, P is
the perimeter of the base, and L is the
slant height
Example 1
Finding the Surface Area of a Pyramid
 Find the surface area of each regular
pyramid
Example 1
Finding the Surface Area of a Pyramid
 Find the surface area of each regular pyramid
S = B + ½PL
Base is a Square
Area of Base = 5(5) = 25
Perimeter of Base
5+5+5+5 = 20
Slant Height = 4
S = 25 + ½(20)(4)
= 25 + 40
= 65 ft2
Example 1
Finding the Surface Area of a Pyramid
 Find the surface area of each regular pyramid
S = B + ½PL
Base is a Hexagon
A=½aP
1
A  (3 3)(36)  54 3
2
Perimeter = 6(6)=36
S = 54 3 + ½(36)(8)
= 54 3 + 144
= 237.5 m2
Slant Height = 8
Cones
 A cone is a solid that has a
circular base and a vertex that is
not in the same plane as the
base
 The lateral surface consists of
all segments that connect the
vertex with point on the edge of
the base
 The altitude, or height, of a cone
is the perpendicular distance
between the vertex and the
plane that contains the base
Right Cone
 A right cone is one in which the vertex
lies directly above the center of the
base
 The slant height of a right cone is the
distance between the vertex and a
point on the edge of the base
Developing the formula for the surface
area of a right cone
 Use the formula for surface
area of a pyramid S = B + ½PL
 As the number of sides on the
base increase it becomes
nearly circular
 Replace ½P (half the perimeter
of the pyramids base) with πr
(half the circumference of the
cone's base)
Theorem 12.5
Surface Area of a Right Cone
 The surface area, S, of a right cone is
S = πr2 + πrL
Where r is the radius of the base and L is
the slant height of the cone
Example 2
Finding the Surface Area of a Right Cone
 Find the surface area of the right cone
Example 2
Finding the Surface Area of a Right Cone
 Find the surface area of the right cone
S = πr2 + πrL
= π(5)2 + π(5)(7)
= 25π + 35π
= 60π or 188.5 in2
Radius = 5
Slant height = 7