Volume and Surface Area Draw a prism Parts of Polyhedron Volume of a Prism Surface Area of a Prism Quiz References

How to draw a Prism 1. Pick a polygon from autoshape. 2. Drag to a size to your liking. 3. Copy and paste. 4. Move the second polygon about 2 cm on top of the first polygon. 5. Draw a line joining 1 vertex of the first polygon to the corresponding vertex of the second polygon. 6. Complete the prism by joining all the corresponding vertices. I feel drawing is a important part of learning spatial learning. If the student can draw it they will have a better idea of the object in 3 D.

Parts of a Prism face edge vertex

Classifying Solids A Polyhedron is a solid that is bounded by polygons, called faces, that enclose a single region of space. An edge is a line segment formed by the intersection of two faces. A vertex of a polyhedron is a point where three or more edges meet. To name a prism or a pyramid, use the shape of the base, for example a pentagonal prism or a triangular pyramid.

Describe each cross section. A cross section is the intersection of a three dimensional figure and a plane. Describe each cross section. The cross section is a triangle The cross section is a circle

A cross section is the intersection of a three dimensional figure and a plane. Describe each cross section

Describe each cross section

A manufacturer of camping gear makes a wall tent in the shape shown in the diagram. a. Classify the three-dimensional figure that the wall tent forms. b. What shapes make up the faces of the tent? How many of each shape are there? c. Draw a net for the wall tent.

answers a. pentagonal prism b. 1 pentagons and 5 rectangles Net of the Tent

Find the Volume of a Prism

Valentine’s Day is approaching. You have 48 chocolate cubes (1 in Valentine’s Day is approaching. You have 48 chocolate cubes (1 in. x 1 in. x 1 in.). How many ways are there to pack these 48 chocolates in the shape of a rectangular box? What are some possible sets of dimensions? I am trying to connect volume to something the students know. That’s why I choose chocolate cubes. To make it easier for the students I am using unit cubes so that the number of cubes equal the volume of the prism. The questions ask are essential questions to get the students to start thinking about dimensions and ways of getting the same volume with different dimensions. 11

How many cubes are in the prism? Make a one-layer rectangular prism that is 4 cubes long and 3 cubes wide. The prism will be 4 units by 3 units by 1 unit. How many cubes are in the prism? Click here to Check your answer It will be nice we can start off with 48 cubes to represent the chocolates cubes. If the students can play with them to arrange these cubes into rectangular prisms with different dimensions. As an example I am using a base of 3 x 4 cubes. 3 x 4 x 1 = 12 12

Add a second layer to your prism to make a prism 4 units by 3 units by 2 units. How many cubes are in the prism? Try again a. 36 cubes b. 12 cubes Try again c. 24 cubes Correct Interaction: Create a interaction where the second then 3rd and 4th level can be added when the students click on the interaction button. When the students click on a certain action buttons the cubes will fly in and form the 2nd, 3rd, and 4th layers. 13

Add a third and fourth layer to your prism to make a prism 4 units by 3 units by 3 units and 4 x 3 x 4. How many cubes are in the prism? Click here to Check your answer 3 layers: 3 x 4 x 3 = 36 4 layers: 3 x 4 x 4 = 48 Interaction: Create a interaction where the second then 3rd and 4th level can be added when the students click on the interaction button. 14

One way to pack the 48 chocolate cubes is 4 x 3 x 4. Is there another way to pack the 48 chocolate cubes? Students experiment with the 48 cubes to derive different dimension. Start off with 48 cubes, drag them to form a rectangular base of different dimensions until all the 48 cubes are done. Then write down the dimensions. 15

Notice that Base Area x Height is the Volume formula Notice that Base Area x Height is the Volume formula. Volume = Base Area x Height

How many cubes would be in the prism with base of 4 units by 3 units if there were 10 layers? The volume will be 4 x 3 x 10 = 120 cubes. The volume of a prism is equal to the length x width x height.

Now we learned that the volume of a rectangular prism is obtained by multiplying the Base Area with the Height. What happened if the prism change to a cylinder? How will you go about finding the volume of the cylinder?

What is the shape of the base? What is the area of a circular base? Examine the cylinder. What is the shape of the base? What is the area of a circular base? Then what will be the volume of a cylinder? Radius height I am trying to connect the volume of a prism to volume of a cylinder. The students always complain that there are too many formula to remember, now they need to remember only 1 formula i.e. volume = Area of the Base x height of the object. As long as the student is able to recognize the shape of the base and the corresponding area formula they can find the volume by multiplying the area with the height. 19

Then what will be the volume of a cylinder? Remember the volume of a prism is Area of the Base x height of the prism Now, consider the cylinder, the area of the cylinder base is r2, what is the volume of a cylinder?  r2 h 2 r 2 r h

You have a cylindrical container with a radius of 2 in You have a cylindrical container with a radius of 2 in. at home and you decided to melt the 48 chocolate cubes and pour the melted chocolate into the cylinder. How high will the chocolate be? This is an extension of the volume formula. If the student know the volume the student will be able to find the height or Base Area of the object. 21

Investigate the Volume of a pyramid You know how to find the volume of a prism. Now we are going to investigate the volume of a pyramid. Consider a prism of size 4 x 4 x 4 in. and a pyramid of the same base and height.

Imagine you filled the inverted pyramid with water and pour the water into the prism. The animation crew will show the animation of pouring the first pyramid full of water into the prism. 25

Notice that 3 pyramids full of water will fill the prism. This will show the animation of pouring the second and third pyramid full of water into the same prism. 26

Observation and Conclusion From the above observation, what can you deduce concerning the volume of the pyramid compare to the prism. The volume of the pyramid is how many times the volume of the prism. Remember the volume of a prism is Length x width x height Hence the volume of pyramid is  1/3 length x width x height.

Homework: Page : 701 # 2 – 10 even Page : 709 # 2 – 10 even

Surface Area of a Prism

Now that the chocolates are packed in the shape of a rectangular prism, he has to wrap the package before sending it out. To keep expenses as low as possible he wonders which sets of dimensions will require the least amount of packaging "Won't it be the same for each? It's still 48 cubes for each set of dimensions. To introduce the concept of surface area to the students, I decided to use the idea of wrapping the prism as a present. 30

The amount of wrapping paper require to wrap a 2 x 4 x 6 prism will be: Consider the net (cut out) of the prism Find the area of each face of the prism.

The sum of all the face areas of the prism is 12 + 24 + 8 + 12 + 24 + 8 = 44

The amount of wrapping paper require to wrap a 2 x 3 x 8 prism will be: Check Answer 2[(8 x 3) + (8 x 2) + (2 x 3)] = 2 [24 + 16 + 6] = 2 x 46 = 92 cm2

The amount of wrapping paper require to wrap a 3 x 4 x 4 prism will be: Check Answer 2[(4 x 4) + (4 x 3) + (4 x 3)] = 2 [16 + 12 + 12] = 2 x 40 = 80 cm2

Conclusion The dimensions that will require the least amount of packaging is the 2 x 4 x 6 prism. The surface area of a prism is the total area of the net.

Homework Page 734 # 10 – 15 Surface area # 16 – 21 volume

Credits and References Agilemind Geometry textbook from Holt, Rinehart and Winston Geometry textbook from Prentice Halls http://www.math.com/tables/geometry/surfareas.htm#prism http://www.shodor.org/interactivate/media/worksheets/463.pdf