Surface Area and
Volume Chapter 12
Exploring Solids
Polyhedron:
a solid that is bounded by polygons, called
faces, that enclose a single region of space.
Edge:
the line segment formed by the intersection of
two faces.
Vertex:
a point where 3 or more edges meet.
Regular:
all faces are congruent regular polygons
Anatomy of a Polyhedron
Examples of Polyhedrons
Example 1. How many
faces? Edges? Vertices?
Types of Solids
Prism= polyhedron
Pyramid=polyhedron
Cone= not a
polyhedron
Types of Solids
Cylinder = not a polyhedron
Sphere= not a polyhedron
Naming Pyramids or Prisms
Name a prism or a pyramid, use
the shape of the base
Bases
Euler’s Theorem
The number of faces (F), vertices
(V), and edges (E) of a
polyhedron are related by the
formula
F + V = E + 2
Convex vs. Concave
Convex = a polyhedron with two points on its
surface are connected by a segment that lies
entirely inside or outside (but along the side) of the
polyhedron
Concave (nonconvex)= a polyhedron with the
segment that goes outside of the polyhedron
Regular Platonic Solids
Regular Tetrahedron
4 faces, 4 vertices, 6 edges
Cube
6 faces, 8 vertices, 12 edges
Regular Octahedron
8 faces, 6 vertices, 12 edges
Regular Dodecahedron
12 faces, 20 vertices, 30 edges
Regular Icosahedron
20 faces, 12 vertices, 30 edges
Cross Sections
If a plane slices through a solid, it forms a cross section.
The cross section of the sphere and the plane is a circle.
Cross Sections
What shape is formed by the
intersection of the plane and the solid?
Square
Cross Sections
What shape is formed by the intersection
of the plane and the solid?
Trapezoid
Stop
Surface Area of Polyhedrons
Surface Area of a Polyhedron
The sum of the areas of its faces
Lateral Area of a Polyhedron
Lateral faces are the parallelograms
formed by connected
corresponding vertices of the bases
The sum of the areas of its lateral
faces
Surface Area and Volume of
Prisms
Prism:
polyhedron with 2 congruent faces,
called bases, that lie in parallel planes.
Lateral faces:
parallelograms formed by connecting
the bases.
Lateral edges:
the segments connecting the vertices
of the bases.
Anatomy of a Prism
More Vocabulary Right prism:
each lateral edge is perpendicular to
both bases
Oblique prisms:
ALL prisms that are not right prisms
Net of a Prism with Triangle Base
Surface Area of a Right Prism
Theorem
The surface area (S) of a right prism can
be found using the formula
S = 2B + Ph = aP + Ph
where B is the area of a base, P is the
perimeter of a base, a is the apothem of
the base, and h is the height or distance
between bases.
Example 1. Find Surface Area
Find the surface area of a right rectangular
prism with a height of 8in., a length of 3in., and a
width of 5in.
S = 2 (15) + 16 (8)
S= 30 + 128
S = 158 in2
5
8
3
Example 2.
Find the surface area of the right prism.
7
7 7
5
S = 2(21.4) + (21)(5)
S = 42.8 + 105
S = 147.8 sq. units
S = 2B + Ph = aP + Ph
S = 2(21.4) + (21)(6.1)
S = 42.8 + 128.1
S = 170.9 sq. units
Cylinders
Cylinder
a solid with congruent
circular bases that lie in
parallel planes.
Lateral area of a cylinder is
the area of its curved
surface.
Surface area is the sum of
the lateral area and the
areas of the two bases.
Right Cylinder
The segment joining the centers of the
bases is perpendicular to the bases
forming a right angle
Net of a Cylinder
Surface Area of a Right
Cylinder Theorem
The surface area (S) of a right cylinder
can be found using the formula
S = 2B + Ch = 2πr2 + 2πrh
where B is the area of a base, C is the
circumference of a base, r is the radius of
the base, and h is the height.
Example 3.
Find the surface area of the right
cylinder.
S = 2B + Ch = 2πr2 + 2πrh
S = 2π (10cm)2 + 2π (10) (30)
S= 628 + 1884
S = 2512 cm2
Example 4. Find the height of a cylinder with a radius of 6.5
and a surface area of 592.19.
S = 2B + Ch = 2πr2 + 2πrh
592.19 = 2π (6.5)2 + 2π (6.5) (x)
592.19= 265.33 + 40.82 (x)
326.86 ≈ 40.82(x)
X ≈ 8
Surface Area of Pyramids
& Cones
Surface Area of Pyramids
Pyramid
a polyhedron with a polygon base and
triangular lateral faces with a common
vertex, aka vertex of the pyramid!
A regular pyramid has a regular polygon
for a base and its height meets the base
at its center.
Anatomy of a Pyramid
Slant Height is the height of a lateral
face of a regular pyramid
Net of a Pyramid
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.
Example 1.
Find the surface area of the regular
pyramid.
S = B + ½ PL
S = 15.6 + ½ (18) (10)
S = 15.6 + 90
S = 105.6 cm2
Example 2.
Cone
Cone
Has a circular base and a
vertex that is not on the same
plane as the base
The radius of the base IS the
radius of the cone
Right Cone
Right Cone
The segment joining the vertex and the center
of the base are perpendicular
Slant height is the distance between the vertex
and the point on the base edge
Lateral surface consists of all segments that
connect the vertex with points on the base
edge
Anatomy of a Cone
Net of a Cone
Surface Area of a Cone
The surface area S of a right
cone is
S = pr2 + prL
where r is the radius of the base
and L is the slant height.
Example 3
Find the surface area of the cone.
S = pr2 + prL
S = π (4) + π(4)(6)
S = 12.56 + 75.36
S = 87.92 in2
S = pr2 + prL
S = π (4)2 + π(4)(6)
S = 50.24 + 75.36
S ≈ 125.60 in2
Example 4 What is the
surface area of the cone?
S = pr2 + prL
S = π (3)2 + π(3)(9.5)
S = 28.26 + 89.49
S = 117.75 in2 or 37.5π
32 + 92 = L2
9 + 81 = L2
L = 9.5
Volume of Solids
What is Volume?
Volume of a solid is the number of cubic
units in its interior.
Measured in cubic units such as cm3
Volume of a Solid Volume of a Cube: V = s3
Volume Congruence
Volume Congruence Postulate:
If 2 polyhedra are congruent then
their volumes are the same.
Volume Addition
Volume Addition Postulate:
The volume of a solid is the sum of the
volumes of all its non-overlapping
parts.
Cavalieri’s Principle IF 2 solids have the same height and the
same cross-section area at every level, then
they have the same volume.
Volume of a Prism
The volume (V) of a prism is
V = Bh
where B is the area of a base and h is the
height.
Example 1.
Find the volume of the prism.
V=Bh
B= ½ (8)(7)
V= 28 (13)
V= 364 cubic units
Volume of a Cylinder
The volume V of a cylinder is
V = Bh = pr2h
where B is the area of a base, h is
the height, and r is the radius of a
base.
Example 2.
Find the volume of the right
cylinder.
V=Bh=πr2h
V=3.14 (3)25
V=3.14 (9)(5)
V≈141.30 cubic units
Example 3
Find the volume of the solid.
Vsmall= 40(20)(10) = 8000
Vlarge= (80)(10)(20) = 16000
Vsmall + Vlarge = 24000 mm3
Example 4.
Find the volume of the
solid.
Vlg= (7.8)(12.4) (9)
Vlg = 870.48 m3
Vcenter= (1.8)(3)(9)
Vcenter= 48.6 m3
Vlg – vcenter = 821.88 m3
Volume of a Pyramid
Volume of a pyramid:
V= 1/3Bh
B is the area of the base and h is the
height
Volume of a Cone
The volume V of a cone is
V = 1/3 Bh = 1/3pr2h,
where B is the area of the base, h is
the height, and r is the radius of the
base.
Example 5
Find the volume of the Cone.
V=1/3πr2h
V = 1/3 (3.14)(62)(5)
V= 334.93 cm3
Example 6
Find the volume of the solid. Assume bottom is a cube
Vcube = (5)(5)(5) = 125 cm3
Vpyramid = (6)(25 cm2)= 150 cm2
Vcube + Vpyramid = 275 cm3
Example 7
Find the volume of the cone inside the cube.
Radius of the base of cone = 10/2 = 5 cm
Cone Height (h) = Side of cube = 10 cm
Volume of the required cone = 1/3πr2h
= 1/3 × 3.14 × 52 × 10
= 1/3 × 3.14 × 25 × 10
= 259.05 cm3
Stop
12.6 Volume and Surface
area of Spheres
Surface Area of a Sphere: S = 4pr2
Surface Area = S ; radius = r
Surface Area
Plane intersects a sphere and the intersection contains
the center of the sphere, the intersection is a great
circle.
Great circles divide the sphere into two hemispheres.
The equator is a great circle.
Using a Great Circle
C = 13.8p ft. for the great circle of a sphere. What is
the surface area of the sphere?
12.6
Volume of a Sphere: V = 4/3 pr3
V = volume ; r = radius
Examples
Radius of
Sphere
Circum. Of
great circle
SA of
sphere
Volume of
Sphere
7mm ? ? ?
? ? 144p in2 ?
? 10p cm ? ?
? ? ? 4000p m3
3
12.7 Similar Solids
Two solids with equal ratios of corresponding linear
measures are called similar solids.
Similar Solids Theorem
If 2 similar solids have a scale factor of a:b, then
corresponding areas have a ratio of a2:b2, and
corresponding volumes have a ratio of a3:b3.
Examples
The prisms are similar with a scale factor of 1:3. Find
the surface area and volume of G if the surface are of
F is 24 ft2 and the volume is 7 ft3.
F
G
Examples
Write the ratio of the two volumes.
V = 512 m3 V = 1728 m3