Lesson 16Distinguishing Between Surface Area and Volume

Learning Goal

Let’s contrast surface area and volume.

Learning Targets

  • I can explain how it is possible for two polyhedra to have the same surface area but different volumes, or to have different surface areas but the same volume.

  • I know how one-, two-, and three-dimensional measurements and units are different.

Lesson Terms

  • base (of a prism or pyramid)
  • polyhedron
  • prism
  • pyramid
  • surface area
  • volume

Warm Up: Attributes and Their Measures

Problem 1

For each quantity, choose one or more appropriate units of measurement.

For the last two rows, think of a quantity that could be appropriately measured with the given units.

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)

Quantities:

  1. Perimeter of a parking lot:

  2. Volume of a semi truck:

  3. Surface area of a refrigerator:

  4. Length of an eyelash:

  5. Area of a state:

  6. Volume of an ocean:

  7. : miles

  8. : cubic meters

Activity 1: Building with 8 Cubes

Problem 1

This applet has 16 cubes in its hidden stack. Build two different shapes using 8 cubes for each.

open applet in presentation mode

For each shape, determine the following information and write it on a sticky note.

  • Give your design a name or a label (e.g., Mae’s First Shape or Eric’s Steps).

  • Determine its volume.

  • Determine its surface area.

Print Version

Your teacher will give you 16 cubes. Build two different shapes using 8 cubes for each. For each shape:

  • Give a name or a label (e.g., Mae’s First Shape or Eric’s Steps).

  • Determine the volume.

  • Determine the surface area.

  • Record the name, volume, and surface area on a sticky note.

Activity 2: Comparing Prisms Without Building Them

Problem 1

Three rectangular prisms each have a height of 1 cm.

  • Prism has a base that is 1 cm by 11 cm.

  • Prism has a base that is 2 cm by 7 cm.

  • Prism has a base that is 3 cm by 5 cm.

  1. Find the surface area and volume of each prism. Use the dot paper to draw the prisms, if needed.

    Isometric grid

  2. Analyze the volumes and surface areas of the prisms. What do you notice? Write 1–2 observations about them.

Are you ready for more?

Problem 1

Can you find more examples of prisms that have the same surface areas but different volumes? How many can you find?

Lesson Summary

Length is a one-dimensional attribute of a geometric figure. We measure lengths using units like millimeters, centimeters, meters, kilometers, inches, feet, yards, and miles.

A drawing of a pencil with an unmarked ruler below it.

Area is a two-dimensional 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.

A drawing of a door with an unmarkee ruler below it and vertically on the side

Volume is a three-dimensional 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.

A drawing of 3 boxes stack on each other with drawings of unmarked rulers along the 3 sides.

Surface area and volume are different attributes of three-dimensional figures. Surface area is a two-dimensional measure, while volume is a three-dimensional 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.

2 figures each made up of 4 cubes. One figure mades a square and the other are end-to-end making a line

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.

5 cubes attached to make a line. 6 cubes attached to make a rectangle shape 2x3.