Surface Area And Volume Of Pyramids And Cones

Today’s Learning Goal We will continue to think about how to find the surface area of different 3-D shapes. We will learn to relate the volume of a pyramid (or cone) with the volume of another 3-D shape that we have already discussed.

Pyramids and Cones Many common and important 3-dimensional objects are not shaped like prisms or cylinders. For example, ice cream is often served in cones. The Egyptian pyramids are examples of pyramids.

Definitions A pyramid is a 3-D figure that has a polygon for a base and triangular sides connecting the base to a point above the base (called the apex). Like prisms, pyramids are named for the shape of its base. Can you name the pyramids below? Triangular pyramid Square pyramid Pentagonal pyramid Hexagonal pyramid

Pyramids and Cones Below are the pyramids we just talked about. If we continued making the base with more and more sides, what will the base eventually look like? Right…we will have a circle for the base.

Surface Area of Pyramids For each of the pyramids below, what would the flat pattern look like? (Draw them in below) So, how would we find the surface area of the pyramids? Right…find the areas of each face and add them all together.

Volumes of Pyramids There is an interesting relationship between the volume of a pyramid and the volume of the prism that surrounds it. Let’s look at a few examples. Consider the following triangular pyramid. What prism would surround it? (Draw it in)

Volumes of Pyramids In the front of the room, I have a triangular pyramid and a triangular prism with the same base and height. The triangular prism is filled with sand. So, the volume of the triangular prism is measured with the amount of sand that it can hold. I am going to pour some of the sand into the triangular pyramid. What portion of the prism’s volume is the pyramid’s volume? Yes…it looks like the volume of the pyramid is 1/3 the volume of the prism that surrounds it!

Volumes of Pyramids In the front of the room, I have a square pyramid and a square prism with the same base and height. Again, the square prism is filled with sand. So, the volume of the square prism is measured with the amount of sand that it can hold. I am going to pour some of the sand into the square pyramid. What portion of the prism’s volume is the pyramid’s volume? Yes…it looks like the volume of the pyramid is 1/3 the volume of the prism that surrounds it!

Volumes of Cones For a cone, what is the thing that would surround it? What do you think is the relationship between the volume of the cone and the cylinder that surrounds it? Yes…based on what we saw with the pyramids, it would be nice if the volume of the cone was 1/3 the volume of the cylinder that surrounds it. Let’s see… Exactly…a cylinder will surround a cone.

Surface Area of Cones Wow! The volume of a pyramid or a cone is 1/3 the volume of the prism or cylinder that surrounds it. The surface area of a pyramid is just all of the areas of the faces added together. What are the different parts of a cone? Yes…the circle at the bottom and then that thing that makes the top part (or bottom part) of the cone. That thing that makes the top part (or bottom part) of the cone is actually part of a circle.

Surface Area of Cones Draw a circle with a compass. Draw two radii. Cut out the pie piece. Fold the remaining part of the circle into a cone. You get two cones from one circle (one from the small pie piece and one from the big pie piece). Try this for several different circles. What makes a tall cone versus a short cone? Right…the smaller the pie piece used, the taller the cone.

Example of a Cone Suppose that when we made our cone, the dimensions of the cone were as follows ( l denotes the slant height of the cone): What would the flat pattern look like for a cone? l = 13 cm r = 5cm 5 cm

Example of a Cone When we made our cones, the slant height of the cone is the same as what part of the circle used to make the cone? Right…the slant height is the same as the radius of the circle. l = 13 cm r = 5cm 5 cm 13 cm The circular base of the cone is part of its surface area. How do we find the area of the circle? Good…  (5) 2  78.5 cm cm 2

Example of a Cone Now we have to determine the area of the curved top part of the cone. The curved top part of the cone is the same as what part of the flat pattern? l = 13 cm r = 5cm 5 cm 13 cm 78.5 cm 2 Yes…the curved top part of the cone is the same as the “pac-man” part of the circle used to make the cone. If we can figure out the part of the circle that was used to make the cone, then we can determine the area of the “pac-man” part of the circle.

Example of a Cone To determine the part of the circle that was used, we can look at the following fraction: l = 13 cm r = 5cm 5 cm 13 cm 78.5 cm 2 The perimeter of the “pac-man” part of the circle is the same as what part of the cone? Yes…the perimeter of “pac-man” is the same as the perimeter of the circle at the base of the cone.

Example of a Cone So, what is the perimeter of the “pac-man” part of the circle? l = 13 cm r = 5cm 5 cm 13 cm 78.5 cm 2 What is the perimeter of the entire circle within the flat pattern? Nice…26   81.64cm Great…10   31.4cm What is the part of the circle that we used to make the cone? Excellent…

Example of a Cone Now we know that we used about.385 of the entire circle in the flat pattern to make the top part of the cone. How would we find the area of the “pac-man” part of the circle? l = 13 cm r = 5cm 5 cm 13 cm 78.5 cm 2 Good…take.385 of the area of the whole circle in the flat pattern. So, we have Area whole =  (13) 2  cm 2 Area pac-man =.385 * = cm cm 2

Example of a Cone We have the area of the circle at the bottom of the cone and the area of the part of the circle that was used to make the curved top part of the cone. Now, what do we do to find the surface area of the cone? 5 cm 13 cm 78.5 cm 2 Beautiful…add 78.5 and together to get cm 2 as the surface area of the cone cm 2

General Formula for Surface Area of Cones For cones, the radius of the circle at the bottom and the slant height, denoted with an l, are the usual dimensions that are given. l r One part of the surface area of the cone we said is the bottom circle. So, what would be one part of the formula for finding the area of the cone? Right…  r 2 is the area of the bottom circle.

General Formula for Surface Area of Cones The radius of the circle becomes what dimension on the cone? Yes…the slant height. The curved top part of the cone is the same as what for the circle? Right…it’s the portion of the circle that you used to make the cone. Now, all we need to figure out is the area of the curved piece at the top of the cone. l l

General Formula for Surface Area of Cones If we can figure out the area of the part of the circle we used to make the cone, then we have the area of the curved top part of the cone. What is the area of the entire circle? l l l2l2 So, a portion of  l 2 is the part that we used to make the curved top part of the cone.

General Formula for Surface Area of Cones What’s the perimeter of the entire circle at the right? l l Right…  *2 l What’s the perimeter of the circle at the bottom of the cone (the part of the circle we used)? The perimeter of the part of the circle we used is the same as what part of the cone? Yes…it’s the same as the perimeter of the circle at the bottom of the cone. Right…  *2r r

General Formula for Surface Area of Cones So, the perimeter of the entire circle is  *2 l and the perimeter of the part of the circle we used to make the cone is  *2r. How do we find out what portion of the circle we used? l l Right…you do the part over the whole. So, we have that we used the following part of the circle:

General Formula for Surface Area of Cones The area of the entire circle we said was  l 2. We only used (part) of the circle to make the cone. So, what would the area of the curved part at the top of the cone be? l l Yes…

General Formula for Surface Area of Cones So, for a cone, the surface area turns out to be: SA cone =  r 2 +  r l l r Area of circle on the bottom Area of curved part on top

Partner Work You have 20 minutes to work on the following questions with your partner.

For those that finish early Determine the VOLUME and SURFACE AREA of the following figure: 4 ft r = 3 ft h = 3 ft

Big Ideas from Today’s Lesson The volume of any pyramid is 1/3 the volume of the prism that surrounds it. The volume of any cone is 1/3 the volume of the cylinder that surrounds it. The surface area of any pyramid is the area of all of the faces added together. The surface area of any cone is found by the following formula: SA cone =  r 2 +  r l Area of circle on the bottom Area of curved part on top

Homework Pgs. 570 – 572 (5, 6, 9, 12 – 14, 20, 22, 24 – 26) Pgs. 580 – 582 (11, 12, 14, 15, 16, 20, 21)