1. Surface Areas of Pyramids
Find the lateral areas of regular pyramids.
Find surface areas of regular pyramids.
Pyramid at the entrance to the Louvre, Paris.

2. LATERAL AREAS OF REGULAR PYRAMIDS
Pyramids have the following characteristics:
All of the faces, except the base, intersect at one point called the vertex.
The base is always a polygon.
The faces that intersect the vertex are called lateral faces and form triangles. The edges of the lateral faces that have the vertex as an endpoint are called lateral edges.
The altitude is the segment from the vertex perpendicular to the base.
Lateral Face
Vertex
Lateral Edge
Altitude
Base

3. LATERAL AREAS OF REGULAR PYRAMIDS
Lateral Face
Vertex
Slant Height
Altitude
Lateral Edge
Altitude
Base (a regular polygon)
Base
Square Pyramid
Regular Square Pyramid

4. LATERAL AREAS OF REGULAR PYRAMIDS
All of the faces of a regular pyramid are congruent isosceles triangles.
The height of each lateral face is called the slant height l of the pyramid
Slant Height
Altitude l
Base (a regular polygon)
Regular Square Pyramid

5. LATERAL AREAS OF REGULAR PYRAMIDS
The figure below is a regular hexagonal pyramid. The lateral area L can be found by adding the areas of all of its congruent triangular faces as shown in the net. l l l l l l s s s s s s l s s s s s s

6. LATERAL AREAS OF REGULAR PYRAMIDS
L = ½sl + ½sl + ½sl + ½sl + ½sl + ½sl
= ½l (s + s + s + s + s + s)
= ½Pl

7. Key Concept Lateral Area of a Regular Pyramid
If a regular pyramid has a lateral area of L square units, a slant height of l units, and its base has a perimeter of P units, then
L = ½Pl l

8. Example 1 Use Lateral Area to Solve a Problem
BIRDHOUSES The roof of a birdhouse is a regular hexagonal pyramid. The base of the pyramid has sides of 4 inches, and the slant height of the roof is 12 inches. If the roof is made of copper, find the amount of copper used for the roof.
L = ½Pl
= ½(24)(12)
= 144 12 4

9. SURFACE AREAS OF REGULAR PYRAMIDS
The surface area of a regular pyramid is the sum of the lateral area and the area of the base.

10. Key Concept Surface Area of a Regular Pyramid
If a regular pyramid has a surface area of T square units, a slant height of l units, and its base has a perimeter of P units and an area of B square units, then
T = ½Pl + B l

11. Example 2 Surface Area of a Square Pyramid24 l 18

12. Example 2 Surface Area of a Pentagonal Pyramid
17 in.
15 in.