Download 12.3 Surface Area of Pyramids and Cones

Warm Up
Find the surface area of the right prisms.
𝑆 = 2𝐵 + 𝑃ℎ
12.3 Surface Area of Pyramids
and Cones
Goal: Find the surface areas of
pyramids and cones.
What is a Pyramid
A Pyramid is a polyhedron
in which the base is a
polygon and the lateral
faces are triangles with a
common vertex (called the
vertex of the pyramid).
Vertex
Regular Pyramids
A Regular Pyramid has a
regular polygon for a base.
The altitude of a regular
pyramid is perpendicular to
the base.
The Slant Height (l) of a
pyramid is the height of one
of the triangles that form the
lateral faces of the pyramid.
THEOREM 12.4: SURFACE AREA
OF A REGULAR PYRAMID
The surface area S of a regular pyramid is
1
𝑆 = 𝐵 + 𝑃𝑙
2
where B is the area of the base, P is the
perimeter of the base, and l is the slant
height
Example: Find the surface area of a
pyramid
1
𝑆 = 𝐵 + 𝑃𝑙
2
The base is a regular hexagon
1
so: 𝐵 = ∗ 𝑠 ∗ 𝑎 ∗ 𝑛
2
1
2
𝐵 = ∗ 8 ∗ 4 3 ∗ 6 ≈166.28
𝑃 = 𝑠 ∗ 𝑛 = 8 ∗ 6 = 48
𝑙 = 18
1
𝑆 = 166.28 + (48)(18)
2
𝑆 = 598.28
Cones
• A cone has a circular base
and a vertex not in the same
plane as the circle
• In a right cone the height
that joins the vertex to the
center of the base is
perpendicular to the base
• The lateral surface of a cone
consists of all segments that
connect the vertex to the
points on the edge of the
base.
THEOREM 12.5: SURFACE AREA OF A
RIGHT CONE
The surface area S of a right
cone is
1
𝑆 = 𝐵 + 𝐶𝑙 = π𝑟 2 + 𝜋𝑟𝑙
2
where B is area of the base,
C is the circumference of the
base,
r is the radius of the base, and
l is the slant height.
Example: Find the surface area of a
right cone.
𝟏
𝑺 = 𝑩 + 𝑪𝒍 = 𝛑𝒓𝟐 + 𝝅𝒓𝒍
𝟐
𝒓=𝟑
𝒍=𝟓
𝐒 = 𝛑𝒓𝟐 + 𝝅𝒓𝒍 = 𝝅𝟑𝟐 + 𝝅(𝟑)(𝟓)
𝑺 = 𝟗𝝅 + 𝟏𝟓𝝅 = 𝟐𝟒𝝅
𝑺 ≈ 𝟕𝟓. 𝟑𝟗𝟖
Homework
Section 12.3 pg. 814 # 3-8, 13-15