Transcript File - Mrs. Hille`s FunZone

Exercise
Find the area of a circle with
a radius of 4.5 units to the
nearest tenth of a square unit
63.6 units2
Exercise
Find the area of a square with
sides of 9.4 units to the
nearest tenth of a square
unit.
88.4 units2
Exercise
How many surfaces would
you have to find the area of in
order to calculate the surface
area for a hexagonal prism?
8
Exercise
How many surfaces would
you have to find the area of in
order to calculate the surface
area for a cylinder?
3
Exercise
How many surfaces would
you have to find the area of in
order to calculate the surface
area for a sphere?
1
Pyramid
A pyramid is a threedimensional figure with a
single polygonal base and
triangular lateral faces that
meet at a common point,
known as the vertex of the
pyramid.
Altitude of a Pyramid
The altitude of a pyramid (H)
is the perpendicular distance
from the vertex to the base.
H
Slant Height
The slant height (l) of a
pyramid is the altitude of
each triangular face.
l
Lateral Surface Area
The lateral surface area (L) of
a pyramid is the sum of the
triangular faces.
Regular Pyramid
A regular pyramid has a
regular polygon as its base,
and its vertex is directly
above the center of the base.
Pyramids with squares or
equilateral triangles for bases
are examples of regular
pyramids.
L = lateral surface area
S = total surface area
B = area of the base
H = altitude
(prism, cylinder, pyramid)
l = slant height
(pyramid, cone)
Formula: Lateral Surface
Area of a Regular Pyramid
1
L = 2 pl The lateral surface
area of a regular
pyramid (L) is equal to
half the product of the
perimeter of the base
(p) and the slant height
of the lateral faces (l).
Formula: Surface Area of
a Regular Pyramid
S = L + B The surface area of a
regular pyramid (S)
is equal to the sum
of the lateral surface
area (L) and the area
of the base (B).
Example 1
Find the lateral surface area
and surface area of the
square pyramid.
1
L = pl
2
13 m
l = 13 m
12 m
p = 4(10) = 40 m
10 m
10 m
1
L = (40)(13)
2
= 260 m2
B = s2 = 102 = 100 m2
S=L+B
= 260 + 100
12 m
2
= 360 m
10 m
13 m
10 m
Example 2
Find the slant height of the
square pyramid.
a = 10 and H = 24
l2 = a2 + H2
= 102 + 242
24
= 100 + 576
= 676
10
l2 = 676 = 26 units 20
Example
What is the surface area of a
pyramid with a square base
with s = 6 units and l = 7
units?
120 units2
Example
What is the slant height of a
pyramid with a square base
with s = 8 units and an altitude
of 3 units?
5 units
Circular Cone
A circular cone is similar to
a pyramid but has a circular
base.
Lateral Surface
The lateral surface is the
curved surface of the cone.
It is not the circular base.
Formula: Lateral Surface
Area of a Circular Cone
1
2
L = cl The lateral surface
= prl area of a circular
cone (L) is equal to half
the product of the
circumference of the base
(c) and the slant height (l).
Substitute for c.
Formula: Surface Area of
a Circular Cone
S = L + B The surface area
= prl + pr2 of a circular
cone (S) is equal to the
sum of the lateral surface
area (L) and the area of
the circular base (B).
Substitute for L and B.
Example 3
Find the surface area of the
cone.
S=L+B
L = prl
= p(3)(10) = 30p + 9p
= 30p
= 39p
B = pr2
= 39(3.14)
10
= p(32)
= 122.5
3
= 9p
Example
What is the slant height of a
circular cone with d = 24
units and an altitude of
5 units?
13 units
Example
What is the surface area of a
circular cone with d = 24
units and an altitude of
5 units?
300p units2 ≈ 942 units2
Sphere
A sphere is a threedimensional closed
surface, every point of
which is equidistant from a
given point called the
center.
Chord
A chord is a line segment
with both endpoints on the
sphere.
Diameter
The diameter is a chord that
passes through the center
of a sphere to an opposite
point on the sphere.
Radius
A radius is a line segment
running from the center of a
sphere to a point on the
sphere.
Plane
A plane intersects a sphere
on a single point or as a
circle.
Great Circle
A great circle is the largest
circle formed by the
intersection of a plane and
a sphere.
Formula: Surface Area of
a Sphere
S = 4A The surface area of a
= 4pr2 sphere (S) is equal to
four times the area of
a great circle (A).
Substitute for A.
Example 4
Find the surface area of the
sphere.
S = 4pr2
= 4p(82)
8 ft.
= 4p(64)
= 256p
= 256(3.14)
≈ 803.8 ft.2
Example
What is the radius of a
circular cone with a surface
area of 75p units2 whose slant
height is equal to its
diameter?
5 units
Example
Determine the length of each
side of the base of a square
pyramid with a surface area
of 343 m2 if the slant height is
three times the length of a
side of the base.
Each side is 7 m.