Lesson 16
Distinguishing Between Surface Area and Volume
Let’s contrast surface area and volume.
16.1: Attributes and Their Measures
For each quantity, choose one or more appropriate units of measurement.
For the last two, think of a quantity that could be appropriately measured with the given units.
Quantities
 Perimeter of a parking lot:
 Volume of a semi truck:
 Surface area of a refrigerator:
 Length of an eyelash:
 Area of a state:
 Volume of an ocean:
 ________________________: miles
 ________________________: cubic meters
Units
 millimeters (mm)
 feet (ft)
 meters (m)
 square inches (sq in)
 square feet (sq ft)
 square miles (sq mi)
 cubic kilometers (cu km)
 cubic yards (cu yd)
16.2: Building with 8 Cubes
This applet has 16 cubes in its hidden stack. Build two different shapes using 8 cubes for each.
For each shape, determine the following information and write it on a sticky note.
 Give a name or a label (e.g., Mae’s First Shape or Eric’s Steps).
 Determine its volume.
 Determine its surface area.
16.3: Comparing Prisms Without Building Them
Three rectangular prisms each have a height of 1 cm.
 Prism A has a base that is 1 cm by 11 cm.
 Prism B has a base that is 2 cm by 7 cm.
 Prism C has a base that is 3 cm by 5 cm.

Find the surface area and volume of each prism. Use the dot paper to draw the prisms, if needed.
 Analyze the volumes and surface areas of the prisms. What do you notice? Write 1 or 2 observations about them.
Can you find more examples of prisms that have the same surface areas but different volumes? How many can you find?
Summary
Length is a onedimensional attribute of a geometric figure. We measure lengths using units like millimeters, centimeters, meters, kilometers, inches, feet, yards, and miles.
Area is a twodimensional attribute. We measure area in square units. For example, a square that is 1 centimeter on each side has an area of 1 square centimeter.
Volume is a threedimensional attribute. We measure volume in cubic units. For example, a cube that is 1 kilometer on each side has a volume of 1 cubic kilometer.
Surface area and volume are different attributes of threedimensional figures. Surface area is a twodimensional measure, while volume is a threedimensional measure.
Two figures can have the same volume but different surface areas. For example:
 A rectangular prism with side lengths of 1 cm, 2 cm, and 2 cm has a volume of 4 cu cm and a surface area of 16 sq cm.
 A rectangular prism with side lengths of 1 cm, 1 cm, and 4 cm has the same volume but a surface area of 18 sq cm.
Similarly, two figures can have the same surface area but different volumes.
 A rectangular prism with side lengths of 1 cm, 1 cm, and 5 cm has a surface area of 22 sq cm and a volume of 5 cu cm.
 A rectangular prism with side lengths of 1 cm, 2 cm, and 3 cm has the same surface area but a volume of 6 cu cm.
Glossary Entries

base (of a prism or pyramid)
The word base can also refer to a face of a polyhedron.
A prism has two identical bases that are parallel. A pyramid has one base.
A prism or pyramid is named for the shape of its base.

face
Each flat side of a polyhedron is called a face. For example, a cube has 6 faces, and they are all squares.

net
A net is a twodimensional figure that can be folded to make a polyhedron.
Here is a net for a cube.

polyhedron
A polyhedron is a closed, threedimensional shape with flat sides. When we have more than one polyhedron, we call them polyhedra.
Here are some drawings of polyhedra.

prism
A prism is a type of polyhedron that has two bases that are identical copies of each other. The bases are connected by rectangles or parallelograms.
Here are some drawings of prisms.

pyramid
A pyramid is a type of polyhedron that has one base. All the other faces are triangles, and they all meet at a single vertex.
Here are some drawings of pyramids.

surface area
The surface area of a polyhedron is the number of square units that covers all the faces of the polyhedron, without any gaps or overlaps.
For example, if the faces of a cube each have an area of 9 cm^{2}, then the surface area of the cube is \(6 \boldcdot 9\), or 54 cm^{2}.

volume
Volume is the number of cubic units that fill a threedimensional region, without any gaps or overlaps.
For example, the volume of this rectangular prism is 60 units^{3}, because it is composed of 3 layers that are each 20 units^{3}.