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