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1.11: Surface Area and Volume

Created by: CK-12

The Polyhedron

Pacing: This lesson should take one class period

Goal: The purpose of this lesson is to introduce students to three-dimensional figures. Polyhedral figures are presented in this lesson and common terms such as edge, vertex, and face are explained.

Grocery Shopping! Begin gathering objects from home that you can use in subsequent lessons. Collect empty cereal, rice, or pasta boxes, empty or full cans of soup, Stackers containers (triangular prisms), etc. These objects will help students visualize nets, presented in the next lesson.

Arts and Crafts Time! Download the nets found on Mathforum’s website. Have students color, cut, and adhere the edges together to form Platonic polyhedra. http://mathforum.org/alejandre/workshops/net.html

For More Information - Click on the following link to gather more information regarding polyhedral figures. http://mathforum.org/alejandre/workshops/unit14.html

Upcoming Vocabulary! Lateral face and lateral edge are two common vocabulary words students should learn. Lateral face is a non-base face (usually the sides). Lateral edge is the segment where two lateral faces meet.

Representing Solids

Pacing: This lesson should take one to two class periods

Goal: The purpose of this lesson is to introduce students to the various types of representations of solid figures. Most will come naturally to students and should be presented as a fun lesson.

Become an Architect! After discussing orthographic views, collect students into groups of 2 or 3. Offer each group a collection of wooden blocks. Their only rule is to build something – a building, house, the Parthenon, etc. Explain to students that architects will often visualize the completed 3-D building from two-dimensional drawings.

Once all the creations are complete, students will rotate to a different structure and sketch its top, front, back, and side views. Students are drawing the orthographic views of a 3-D structure. Students may complete additional drawings as an assignment, in-class activity, or extra credit.

Cross Section View, Using a Breadknife! When discussing cross-sections, bring in a loaf of bread and a breadknife. Illustrate the perpendicular cross by cutting through the bread vertically. You could also show non-perpendicular cross section by cutting through the bread at different angles.

Nets. Using the collected cereal boxes, have students turn these into nets of prisms and draw sketches. Students will use real life objects to visualize nets.

Prisms

Pacing: This lesson should take one to two class periods

Goal: This lesson introduces students to the surface area and volume formulas of prisms.

Flashcards! The focus of these flashcards is to organize 3-dimensional formulas for area and volume. You could use the following chart to help students begin to organize their flashcards.

Surface Area Formula Volume Formula
Cube
Rectangular Prism
Triangular Prism
Hexagonal Prism

Additional Examples:

1. How much honey can 65 honeycomb cells hold if each hexagonal cell is \frac{1}{8}” long by \frac{1}{4}” deep? 2.64\;\mathrm{in}^3

http://www.flickr.com/photos/justusthane/1252907196/

Cylinders

Pacing: This lesson should take one to two class periods

Goal: This lesson introduces students to the surface area and volume formulas of cylinders.

Organization! Add the following rows to your table began in the previous lesson.

Surface Area Formula Volume Formula
Cylinder

Visualization! Using soup cans or other cylindrical objects, show students the lateral face of a cylinder by peeling the label from the can. Students will see the lateral face is a rectangle, not a circle.

Additional Example:

  1. A drinking straw has is 11” long with a 0.5” diameter. How much plastic is needed to form the straw? 17.28\;\mathrm{in}^2.
  2. Using the same straw, how much liquid can it hold? 2.16\;\mathrm{in}^3

http://www.flickr.com/photos/cedsarlette/3002603565/

Pyramids

Pacing: This lesson should take one to two class periods

Goal: This lesson introduces students to the surface area and volume formulas of cylinders.

Lab Investigation! Before you read through the volume of a pyramid section, have your class complete this lab! This is a great way to demonstrate the relationship between the volume of a prism and the volume of a pyramid.

Fill a gallon jug with water colored with food coloring. Separate students into groups of three or four. Each group should receive a prism and its matching pyramid. The bases and heights must be identical for this to work! Instruct one student to measure the necessary values of the 3-dimensional figures (altitude and lengths of base) while another student records the information. A third student will fill the pyramidal figure with colored water and pour it into the prism. The goal is to determine how many times it will take to fill the prism. The answer should be approximately 3.

Encourage students to write a hypothesis regarding the relationship between these two volumes. Students should state that 3*\mathrm{pyramid} = \mathrm{prism.}

Additional Examples:

  1. Draw a net for a right pentagonal pyramid.

Cones

Pacing: This lesson should take one to two class periods

Goal: This lesson introduces students to the surface area and volume formulas of cones.

Additional Examples:

1. Draw the net for a right cone with diameter 3\;\mathrm{cm} and height 5\;\mathrm{cm.}

2. How much material is needed to make the waffle cone shown below with dimensions 12” tall with 5” diameter? 96.29\;\mathrm{in}^2

http://www.flickr.com/photos/goddess-arts/2886293009/

3. The conical water cup has dimensions 6” with a 3.5” diameter. How much can it hold? 57.73\;\mathrm{in}^3

http://www.flickr.com/photos/63965847@N00/27208445/

Cones

Pacing: This lesson should take one to two class periods

Goal: This lesson introduces students to the surface area and volume formulas of spheres.

Geography Connection! Using a globe, show students how a sphere is formed using rotation. The teardrop shaped pieces are called gores. Once placed together and folded so they meet at the top and bottom (poles), a sphere is formed.

The great circle is a cross section of a sphere cutting through the widest part of the sphere, the equator. Any other cross section is called a small circle. Using the globe, show students examples of each (i.e. The Artic Circle and the Equator).

Extra research! Have students research the different types of maps and list the pros and cons of each, relating to the true topography of Earth.

Additional Examples:

  1. How much leather is needed to make a baseball with a 6.5” diameter? 132.72 \;\mathrm{in}^2
  2. A plant container is a hemisphere with a radius of 17.” How much dirt can it hold? Don’t forget to divide your answer in half – 1286.22\;\mathrm{in}^3

Similar Solids

Pacing: This lesson should take one to two class periods

Goal: This lesson introduces students to the surface area and volume formulas of cones.

Gulliver’s Travels Revisited! In a previous lesson, we connected Gulliver’s Travels to areas of similar figures, such as the footprints of Gullivers versus the Lilliputians. Extend this topic to surface area and volume.

The surface area (amount of material needed to make clothing, etc.) has a ratio \left (\frac{\mathrm{image\ area}} {\mathrm{preimage\ area}}\right ) of k^2. Therefore, if a Lilliputian was \frac{1}{12} the size of Gulliver and Gulliver was \frac{1}{12} the size of a Brobdignagian, the amount of material needed to cloth a Brobdignagian would be 144^2 times the amount needed for a Lilliputian!

This also related to volume (weight). The ratio here would be k^3. Have fun and try lots of different ratios!  

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