Chapter 9: Measurement and the Metric System
Section 9.3: Volume and Surface Area
Total Surface Area of a Cylinder and a Prism
•
Lateral area: the area of the regions bounded by the lateral faces of a prism
•
Total surface area: the area of the lateral faces combined with the area of both bases
•
Areas of a surface: the lateral surface area of a given surface is the sum of the areas of all of the faces
or sides (other than the bases) of the figure. The total surface area is the lateral surface area added to the
area of the bases of the space figure.
•
Surface area of a right rectangular prism: SArectangular prism = 2(lw + lh + wh)
B
B
h
h
B
B
•
Surface area of right regular n-gonal prism: SAn-gonal prism = Ph + aP = nhs + ans = ns(h + a)
B
B
h
h
B
B
•
Surface area of right cylinder: SAcylinder = 2πr2 + 2πrh = 2πr(r + h)
B
B
r
r
h
h
2(pi)r
B
B
Volume Measure
•
Volume: associates a unique number with each closed space region; to find the volume of a prism or
cylinder we multiply base area times height; volume is given in cubic units or units cubed
1. The number of squares in the base area matches the number of cubes or cubic units in the bottom
layer
2. The number of segments in the height matches the number of layers
•
Volume of a right rectangular prism: The volume of a right rectangular prism is the product of the
prism’s length, width, and height. You can also think of it as the area of the base (Arectangle = lw) times
the height, h, then we have – Vrectangular prism = lwh
B
h
B
w
l
•
Properties of Volume:
1. Volume is additive; that is, the volume of the whole is equal to the sum of the volumes of the
nonoverlapping parts. If R and S are nonoverlapping space regions (possibly with surfaces in
common), then the volume of R ∪ S is equal to the volume of R plus the volume of S.
2. If space region R has the same size and the same shape as space region S, then the volume of space
region R equals the volume of space region S.
3. If a space region is cut into parts and reassembled without overlapping to form another space region,
then the space regions have the same volume.
Units of Measure for Volume
•
Volume: cubic linear measures –
o 1728 in3 = 1 ft3
o Where do we talk about cubic feet of a substance in the real world? swimming pool water
o 27 ft3 = 1 yd3
o
Where do we talk about cubic yards of a substance in the real world? concrete
•
Volume of a Cube: Volume of a cube is found by: Vcube = s3 where s is the length of an edge of the
cube or one of the sides of the square faces
•
Cubic centimeter: Sometimes referred to in common language as a cc; 1 cm3 = 1 mL where L is the
metric unit for volume known as liter.
Cavalieri’s Principle
•
Cavalieri’s Principle: Given: a plane and any two solids with bases in the plane. If every plane parallel
to the given plane intersects the two solids in cross-sections that have the same area, then the solids have
the same volume.
Volume of a Prism and a Cylinder
•
Oblique Prism: an oblique prism is any prism that is not a right prism.
•
Oblique Cylinder: an oblique cylinder is any cylinder that is not a right cylinder.
•
Cavalieri’s principle applied: the formulas for finding the volumes of oblique objects remain the same,
but the height is no longer the length of an edge or lateral face. The height is determined by the
perpendicular distance between the planes containing the bases.
B
h
B
w
l
•
Volume of Prism or Cylinder: The volume of any prism or cylinder is the area (B) of a cross-section
cut by a plane parallel to the base multiplied by the altitude or height (h), then Vprism or cylinder = Bh
•
Triangular Prism: Vtriangular prism = Bh = 0.5bh1h2
h1
b
h2
•
Cylinder (tin can): Vcylinder = Bh = πr2h
B
r
h
B
Surface Area and Volume of a Pyramid
•
Regular pyramid: the lateral faces are isosceles triangles
•
Slant height of pyramid: a perpendicular from the vertex of the pyramid to the base of one of the
isosceles triangles is called the slant height (t) of the pyramid.
•
Area of the lateral faces of a regular pyramid: the area of each of the isosceles lateral faces of a
regular pyramid is one-half the slant height (t) times an edge (s) of the polygonal base, then SAone lateral
face
= 0.5ts; the base has n edges, so the surface area of all of the lateral faces is SAall lateral faces = 0.5nst =
0.5Pt since Pregular n-gon = ns
•
Surface Area of a Pyramid: for a regular pyramid with slant height t, perimeter of the base P, and area
of the base B: SAregular pyramid = 0.5Pt + B = 0.5nst + 0.5ans = 0.5ns(t + a)
•
Surface area square pyramid: SA = 0.5Pt + B = 0.5nst + s2 = s(0.5nt + s)
t
s
•
Slant height of cone: a perpendicular from the vertex of the cone to the base of the cone is called the
slant height (t) of the cone.
•
Lateral surface area of a cone: half of the product of the circumference C and the slant height t --SAlateral cone = 0.5tC = 0.5t(2πr) = πrt
NOTE: numbers are written before symbols or variables, symbols are written before variables,
variables are written in alphabetical order by convention.
•
Surface Area Cone: sum of the lateral surface area and the area of the base B: SAcone =
πrt + πr2 = πr(t + r)
t
r
•
Volume of a Pyramid: Vpyramid =
h
s
•
Volume of a Cone: Vcone =
h
r
Volume and Surface Area of a Sphere
•
Surface Area of a Sphere: SAsphere = 4πr2
r
•
Volume of a Sphere: Vsphere =
r
Exercise Sets:
Homework:
p. 463: 1ac, 2, 3, 5, 7, 9, 11, 12, 13, 14ac, 16, 17, 18, 19, 20ac
Blazeview:
p. 463: 15