The surface area of a three-dimensional figure is the area that would be covered if its surface was peeled off and laid flat, measured in square units. The volume is the measure of cubic units within a three-dimensional figure. Formulas are provided to calculate the surface area and volume of boxes, cylinders, cones, spheres, pyramids, and other shapes. Examples demonstrate applying the formulas to real-world applications.

1. Volume and Surface Area
The surface area of a three-dimensional figure is the area that would be
covered if the surface were “peeled” off the figure and laid flat. It is
measured in square units (just like any other area measure).
The volume is the measure of cubic units in a three-dimensional figure.
A cubic unit is a cube that is one unit long in each of the three dimensions.

2. Box
The surface area of a box --- also known as a rectangular parallelepiped ---
is the sum of the areas of the six sides
S 2lh 2lw 2hw
h
w
l
The volume of a box is the product of the three dimensions.
V l w h

3. Right Circular Cylinder
The surface area of a right circular cylinder --- the shape of a soup can or
soda can --- is the sum of the areas of the top, the bottom and the side. Th
area of the side is the height of the cylinder times the distance around the
cylinder.
r
Area of top = area of bottom= S = 2 r2 + 2 rh
r2
h
Area of side =
2 rh
The volume of a cylinder equals the area of the base times the height.
V= r 2h

4. Right Circular Cone
The surface area of a right circular cone --- the shape of a witch’s hat or an ice
cone --- is the sum of the areas of the base of the cone and the side of the con
Area of base = S= r2 + r√r2 + h2
r2
h
Area of side =
r√r2 + h2 r
The volume of a right circular cone is 1/3 times the area of the base times the
V = 1/3 r 2h

5. Sphere
To picture the surface area of a sphere imagine peeling an orange and flatte
the peel on a table. The area covered would be the surface area of that sph
S = 4 r2 Note this is the area of 4 circles with radius r.
r
The formula for the volume of a sphere is
V = 4/ 3 r 3

6. Pyramids
A pyramid is a three-dimensional figure with a polygon as its base and isoscele
triangles as its sides. For any pyramid if B represents the area of its base and
the height of the pyramid its volume equals 1/3 times the area of the base time
the height.
V = 1/3Bh
h
rectangular pentagonal
pyramid pyramid

7. Application
A rectangular office building measures 57 yds. by 36 yds. and is 28 stories
high. If there are 3.5 yds. per story what is the volume of the building?

8. Application
The materiel to make an aluminum soda can costs $0.01
per square inch. How much does it cost to make a soda can
5 in. high and 2.5 in. in diameter?
2.5 in.
What is the volume
5 in. of the can?

9. Application
You work at a local ice cream parlor making waffle cones. If a finished cone
6 in. high and has a base diameter of 4 in. what is the surface area of the con
(not including the area of the base)?
If you fill the cone with soft ice cream topped with a hemisphere of soft ice
cream what is the volume of the ice cream?

10. Application
The Transamerica Tower in San Francisco is a pyramid with a square base
Each side of the base is 52 meters and the height is 260 meters. What is t
volume of the Transamerica Tower?

11. Polyhedrons
A polyhedron is a 3-dimensional figure whose sides or faces are polygo
Regular polyhedrons have faces which are all the same regular polygon
all equilateral triangles
all squares
all regular pentagons
There are only 5 known regular polyhedrons:
Tetrahedron --- 4 sides --- all equilateral triangles
Hexahedron --- 6 sides --- all squares
Octahedron --- 8 sides --- all equilateral triangles
Dodecahedron --- 12 sides --- all regular pentagons
Icosahedron --- 20 sides --- all equilateral triangles

12. Prisms
A prism is a three-dimensional figure that has two of its faces in parallel plan
These faces are congruent polygons. The rest of its faces are parallelogram
formed by connecting (drawing a line between) corresponding vertices of th
two congruent polygons.
triangular trapezoidal
prism prism