This document discusses how to find the surface area of pyramids and cones. It provides formulas for calculating the surface area of regular pyramids and right cones. For a regular pyramid, the surface area is equal to the area of the base plus half the product of the perimeter of the base and the slant height. For a right cone, the surface area is equal to the area of the circular base plus the product of the radius of the base and the slant height. The document includes examples of applying these formulas to find the surface area of different pyramids and cones.

1. 12.4 Surface Area of
Pyramids and Cones
Geometry
NCSCOS: 1.02; 2.03; 2.04

2. Objectives/Assignment
• Find the surface area of a pyramid.
• Find the surface area of a cone.

3. Finding the surface area of a
pyramid
• A pyramid is a polyhedron in which the base is a
polygon and the lateral faces are triangles with a
common vertex. 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 a pyramid is the
perpendicular distance between the base and
the vertex.

4. More on pyramids
• A regular pyramid has a
regular polygon for a
base and its height meets
the base at its center.
The slant height of a
regular pyramid is the
altitude of any lateral
face. A nonregular
pyramid does not have a
slant height.

5. Pyramid Arena

6. Ex. 1: Finding the Area of a Lateral
Face
• Architecture. The lateral faces of the
Pyramid Arena in Memphis, Tennessee,
are covered with steal panels. Use the
diagram of the arena to find the area of
each lateral face of this regular pyramid.

8. Hexagonal Pyramids
• A regular hexagonal
pyramid and its net are
shown at the right. Let b
represent the length of a
base edge, and let l
represent the slant height
of the pyramid. The area
of each lateral face is
1/2bl and the perimeter
of the base if P = 6b. So
the surface area is as
follows:

9. Hexagonal pyramid
S = (Area of base) + 6(Area of lateral face)
S = B + 6( ½ bl)
S = B + (6b)l
S = B + Pl
Substitute
Rewrite 6( ½ bl) as ½ (6b)l.
Substitute P for 6b
Surface Area of a Regular Pyramid
The surface area S of a regular pyramid is:
S = B + ½ Pl, where B is the area of the base, P is
the perimeter of the base, and l is the slant height.

10. Ex. 2: Finding the surface area of a
pyramid
• To find the surface area
of the regular pyramid
shown, start by finding
the area of the base.
• Use the formula for the
area of a regular
polygon,
½ (apothem)(perimeter).
A diagram of the base
is shown to the right.

11. Ex. 2: Finding the surface area of a
pyramid
After substituting, the
area of the base is
½ (3 )(6• 6), or
1 square meters.
√3
√3

12. Surface area
• Now you can find the surface area by
using 54 for the area of the base, B.√3

13. Finding the Surface Area of a Cone
• A circular cone, or cone,
has a circular base and a
vertex that is NOT in the
same plane as the base.
The altitude, or height, is the
perpendicular distance
between the vertex and the
base. In a right cone, the
height meets the base at its
center and the slant height
is the distance between the
vertex and a point on the
base edge.

14. Finding the Surface Area of a Cone
• The lateral surface of a
cone consists of all
segments that connect the
vertex with points on the
base edge. When you cut
along the slant height and
like the cone flat, you get
the net shown at the right.
In the net, the circular base
has an area of πr2
and the
lateral surface area is the
sector of a circle.

15. More on cones . . .
• You can find the area of this sector by
using a proportion, as shown below.
Area of sector
Area of circle
=
Arc length
Circumference
Set up proportion
Area of sector
πl2 =
2πr
2πl
Substitute
Area of sector = πl2
•
2πr
2πl
Multiply each side by πl2
Area of sector = πrl Simplify
The surface area of a cone is the sum of the base
area and the lateral area, πrl.

16. Theorem
• 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

17. Ex. 3: Finding the surface area of a
cone
• To find the surface area
of the right cone shown,
use the formula for the
surface area.
S = πr2
+ πrl Write formula
S = π42
+ π(4)(6) Substitute
S = 16π + 24π Simplify
S = 40π Simplify
The surface area is 40π square inches or about 125.7
square inches.