<meta http-equiv="refresh" content="1; url=/nojavascript/"> Surface Area and Volume | CK-12 Foundation

# 4.11: Surface Area and Volume

Created by: CK-12

## The Polyhedron

I. Section Objectives

• Identify polyhedral.
• Understand the properties of polyhedral.
• Use Euler’s formula to solve problems.
• Identify regular (Platonic) polyhedral.

II. Multiple Intelligences

• Teach the material in this lesson, and then differentiate it by having the students work in groups to test out Euler’s formula.
• Provide the students with actual solids or with diagrams of different polyhedral.
• The students need to use the solids or diagrams to label each polyhedra and to come up with a way to demonstrate how Euler’s formula works.
• Allow students time to choose one solid that they are going to use for their “teaching” session.
• When students are finished preparing, allow time for them to teach the other students in the class how Euler’s formula works.
• They need to be clear on the number of faces, edges and vertices of their solid.
• Allow time for the students to answer questions.
• Provide feedback to the students.
• Intelligences- linguistic, logical- mathematical, visual- spatial, bodily- kinesthetic, interpersonal, intrapersonal

III. Special Needs/Modifications

• Provide students with notes on polyhedrons.
• Polyhedrons
• $3D$
• Made of polygons and only polygons- faces
• Polygons join at the edges.
• Edges meet in points called vertices.
• No gaps between them.
• Review the definition of a polygon.
• Write out Euler’s Formula.

IV. Alternative Assessment

• Pay close attention to the teaching session.
• Do the students prove Euler’s formula or simply state reasons.
• Ask questions and stretch the students to really demonstrate how Euler’s formula works and makes sense.

## Representing Solids

I. Section Objectives

• Identify isometric, orthographic, cross- sectional views of solids.
• Draw isometric, orthographic, cross- sectional views of solids.
• Identify, draw and construct nets for solids.

II. Multiple Intelligences

• Make this lesson very interactive by giving students the following hands- on tasks.
• Students may work in small groups for this activity.
• In the activity, be sure that students have graph paper, dot paper, plain paper, rulers, tape and colored pencils.
• Students are going to choose a solid to work with. You can provide students with a model of a solid if you have them.
• Then students are going to create four different things.
• 1. Create an orthographic projection of their solid.
• 2. Create a cross- section of the solid.
• 3. Create a net for the solid.
• 4. Use the net to create an actual model of the solid.
• Intelligences- linguistic, visual- spatial, bodily- kinesthetic, logical- mathematical, interpersonal, intrapersonal.

III. Special Needs/Modifications

• Review polyhedral.
• Review faces, edges and bases.
• Define isometric.
• Define perspective.
• Define orthographic projection.
• Define cross- section.
• When students draw an orthographic projection, they will need the following views:
• 1. Top
• 2. Left side
• 3. Back
• 4. Right side
• 5. Front
• 6. Bottom

IV. Alternative Assessment

• An assessment of student understanding can easily be completed by looking at the student work.
• Were the students able to create a net that built a solid?
• Did they use the same solid for all of the pieces of the activity?
• Where did students have challenges?

## Prisms

I. Section Objectives

• Use nets to represent prisms.
• Find the surface area of a prism.
• Find the volume of a prism.

II. Multiple Intelligences

• When differentiating this lesson, you want to divide it into two sections. The first section is going to be on surface area and the second section is going to be on volume.
• The first thing to do is to present the information on surface area.
• When you teach surface area, have the students create a net of a prism (on graph paper) to work with.
• Once the students have created their nets, then use the formulas for surface area to show students how to calculate the surface area of the prism that they created.
• When finished, allow time for the students to complete their work.
• Then move on to volume.
• Use the same net to calculate the volume of the solid.
• Have the students see where the formula for finding the volume of a solid comes from.
• Building the lesson in this way connects the last few lessons together. We have connected surface area and volume with polyhedra.
• Allow time for students to share their work in the large class or in small groups.
• Intelligences- linguistic, logical- mathematical, bodily- kinesthetic, interpersonal, intrapersonal

III. Special Needs/Modifications

• Write out all new terms and information on the board. Be sure that students copy this information in their notebooks.
• Define prism. What makes a prism a prism?
• Show students the difference between a right prism and an oblique prism.
• Define Area Congruence Postulate
• Define Surface Area.
• Review formulas for area of a triangle, parallelogram, rectangle and square.
• Show the difference between Lateral area and surface area.
• Define Volume.
• Define Volume Congruence Postulate.
• Show how these two postulates are similar to the ones on surface area.
• Be sure that students know that the capital B in the volume formula means the area of the base not the length of the base.

IV. Alternative Assessment

• Listen to student responses during class discussions.
• More students will have a chance to share their work in small groups.
• If you use small groups, walk around and listen in on each group.
• Make notes of any students who are having difficulties.

## Cylinders

I. Section Objectives

• Find the surface area of cylinders.
• Find the volume of cylinders.
• Find the volume of composite three- dimensional figures.

II. Multiple Intelligences

• To differentiate this lesson, you can make it very hands- on by using some different cylinders. Example- Quaker Oats containers, soda cans, etc.
• Teach the material in the lesson.
• Then have the students work to find the surface area and volume of each cylinder.
• The students will need string to determine circumference, rulers, colored pencils and paper.
• Have students draw a net for their cylinder and label all of the measurements involved.
• Then they complete their work.
• Provide time for students to share their work in small groups.

III. Special Needs/Modifications

• Review finding the area and circumference of a circle.
• Define cylinders.
• Show students the difference between right cylinders and oblique cylinders.
• Write out the formulas for surface area and volume of cylinders.
• Steps to Working with Composite Solids
• 1. Break each composite solid into its smaller solid parts.
• 2. Select the correct formula for either surface area or volume based on the problem.
• 3. Find the surface area or volume of each smaller solid.
• 4. Add/subtract the results of the surface area or volume based on the question.

IV. Alternative Assessment

• Observe students as they work in small groups.
• Have the students share their work from the cylinder activity when finished.
• Check student work for accuracy.
• Provide assistance and feedback when necessary.

## Pyramids

I. Section Objectives

• Identify pyramids.
• Find the surface area of a pyramid using a net or formula.
• Find the volume of a pyramid.

II. Multiple Intelligences

• Differentiate this lesson by having students complete the following activities.
• Students are going to be working with a pyramid of their choosing.
• Begin by dividing students into groups. While the students will be working with different pyramids, they will have the support of the other students in the group as they work.
• Each student is to choose a type of pyramid: triangular, square, pentagonal, hexagonal, etc.
• It is fine if two students in the group choose the same pyramid.
• Next, the students need to draw a net for their pyramid. Have them include measurements of each part of the pyramid.
• Then, students need to find the surface area of the pyramid.
• The lateral area of the pyramid
• The volume of the pyramid
• Have students show all of their work.
• When finished, they need to check the work of one other student in their group.
• Finally, collect student work to assess levels of understanding.
• Intelligences- linguistic, logical- mathematical, visual- spatial, interpersonal, intrapersonal.

III. Special Needs/Modification

• Define regular pyramid.
• Identify types of pyramids according to each base.
• Define slant height.
• Write the formulas for surface area, lateral area and volume on the board.
• Be sure students understand how to find each measurement.

IV. Alternative Assessment

• Collect and review student work.
• Is the net correctly drawn and labeled?
• Is the surface area of each net correct?
• Is the lateral area of each correct?
• Is the volume correct?
• Have there been any corrections to the work of the students by a peer?

## Cones

I. Section Objectives

• Find the surface area of a cone using a net or formula.
• Find the volume of a cone.

II. Multiple Intelligences

• Begin by having the students work to build cones.
• Students begin by drawing a half circle and measure the top of it.
• This top measurement becomes the circumference of the cone.
• Then they cut out the half circle.
• Have the students turn it into a cone.
• Then use the descriptions of the parts of the cone in the text to help the students to understand the parts of the cone.
• After this teach the material in the lesson and then move on to the next activity.
• Next, divide the students into groups.
• In each group, ask the students to choose either the surface area formula or the volume formula.
• With each formula, the students need to find a way to teach how the formula is put together for the other students in the class.
• The students will demonstrate the meaning of each part of the formula and present an example of how to use the formula.
• Intelligences- linguistic, logical- mathematical, visual- spatial, bodily- kinesthetic, interpersonal, intrapersonal.

III. Special Needs/Modifications

• Define cone as a single curved base that tapers to a single point. This point is called the apex.
• Base can be a circle or an oval.
• Right cone- apex in the center
• Oblique cone- apex not in the center
• Write the formulas for surface area and volume on the board.

IV. Alternative Assessment

• Create a checklist of the things that the students should be teaching in their lessons.
• Use these checklists during the presentations.
• Make notes of the things that the students cover.
• Be sure to provide feedback for the things that the students miss in their presentations.

## Spheres

I. Section Objectives

• Find the surface area of a sphere.
• Find the volume of a sphere.

II. Multiple Intelligences

• Begin by showing students some spheres of different sizes. You can use a baseball, a globe and a basketball for example.
• Often the measurement of a ball is given according to the diameter. Have the diameter of any object that you show the students close by. Use one- for example, a fourteen inch basketball to demonstrate the following.
• Parts of a circle
• 1. $O =$ center point
• 2. $r =$ radius ($\frac{1}{2}$ of diameter)
• 3. $d =$ diameter
• 4. Chord- intersects the center of the circle or sphere.
• 5. Secant- line, ray or line segment that intersects in two places and extends OUTSIDE the sphere
• 6. Tangent- intersects the sphere at only one point.
• SA of a Sphere- use the formula and then use the examples to have students work to find the SA of one or more of the given objects.
• For example, find the SA of the 14” basketball.
• Volume of a sphere- use the formula and then use the examples to have students work to find the volume of each given object.
• Allow time for students to share their work.
• Intelligences- linguistic, logical- mathematical, visual- spatial, bodily- kinesthetic, interpersonal, intrapersonal.

III. Special Needs/Modifications

• Review circles.
• Compare circles to spheres.
• Show how the parts of a circle relate to the parts of a sphere.
• Review the meaning of surface area.
• Review the definition of volume.

IV. Alternative Assessment

• Assess student understanding of the material through the discussion and through student answers when working with the given objects.

## Similar Solids

I. Section Objectives

• Find the volumes of solids with bases of equal areas.

II. Multiple Intelligences

• To differentiate this lesson, begin by teaching the content in the lesson.
• Ask the student’s to draw an example of Cavalieri’s Principle (Volume of a solid postulate)
• Have students share their example with a peer and then allow time for student sharing.
• The class participation will give you time to see if the students understand the principle.
• Then move on to working with similar solids. The students are going to draw a pair of similar solids and then work to problem solve with the similar solids.
• Tell students to draw a solid that is similar to a rectangular prism with a depth of $4$, a width of $6$, and a height of $9$.
• Students should begin by drawing this given rectangular prism and then draw one similar to it.
• Once this is similar, ask them to write ratios to demonstrate that the prisms are similar.
• Next, have the students find the surface area of each prism and demonstrate that they are similar through the Similar Solids Postulate.
• Finally, ask students to find the volume of each prism and demonstrate that they are similar through the Similar Solids Postulate.
• Allow time for the students to share their work.
• Intelligences- linguistic, logical- mathematical, visual- spatial, interpersonal, intrapersonal.

III. Special Needs/Modifications

• Review surface area.
• Review volume.
• Review scale factor.
• Write Cavalieri’s Principle on the board. Rename it as the Volume of a Solid Postulate.
• Review similar solids and writing equal ratios.

IV. Alternative Assessment

• Walk around as students work and assess their understanding through observation.
• You can collect student work to use as a classwork grade.
• Offer assistance to students who are in need of help.
• Use flexible grouping to assist these students.

## Date Created:

Feb 22, 2012

Feb 23, 2012
You can only attach files to None which belong to you
If you would like to associate files with this None, please make a copy first.