Surface Area And Volume Of Cylinders

Today’s Learning Goal We will continue to work with flat patterns to help us see the total area of the 3-D shape. We will continue to think about how to find the volume of different looking 3-D objects.

Prisms So far, we have looked at several different prisms. Some of them are drawn below. If we continued making the top and bottom with more and more sides, what will the top and bottom eventually look like? Right…we will have 2 circles for the top and bottom.

Cylinders A cylinder is a 3-dimensional shape with a top and bottom that are identical circles. There are many packages and containers that are shaped like cylinders. For example, salt, juice concentrate, soup, oatmeal, and tuna are often sold in packages shaped like cylinders.

Cylinders Most of you probably have made a cylinder before sometime in your childhood. Do you remember taking a piece of rectangular paper and rolling it up to make a telescope to look through? Your cylindrical telescope is made of 3 pieces: (1) a circular top, (2) an identical circular bottom, and (3) the rectangular piece of paper used to join the two circles.

Cylinders As with prisms, the bottom of a cylinder is called the base, and the distance from the base to the top is called the height. You can describe a cylinder by giving its dimensions. The dimensions of a cylinder are: the radius of its base (or top) and its height. h r

Cylinders If we know the radius and the height of a cylinder, then we can figure out its surface area and volume. How would we figure out the volume of the cylinder if the radius was 3 cm and the height was 10 cm? Right…figure out how many unit cubes would fit on the bottom layer and then multiply by 10 (because there would be 10 identical layers). 3 cm 10 cm

Cylinders How would we figure out how many unit cubes could fit on the bottom layer? So, if cubic centimeters fit in the bottom layer, then how many would fit in the entire cylinder? Right…determine the area of the circle. So,  (3) 2  28.27cm 3 fit in the bottom layer. 3 cm 10 cm Exactly…10*28.27 = 282.7cm 3 because there are cubes in each of 10 identical layers.

Cylinders The surface area of any shape is the area of all of its pieces put together. We have been drawing flat patterns to help us see the different pieces. What would the flat pattern of a cylinder look like?

Cylinders If the radius of the cylinder was 3 cm and the height was 10 cm, then the area of the circles is  (3) 2  cm 2 each. Two of them would be 2*28.27 = cm 2. 3 cm 10 cm cm 2

Cylinders To find the area of the rectangular middle, we need to know its base length and height. The height of the rectangle is the same as what dimension of the cylinder? 3 cm 10 cm Yes…the height cm 2 10 cm

Cylinders The base length of the rectangle is the same as what on the cylinder? 3 cm 10 cm Right…the circumference of the circle cm 2 10 cm

Cylinders 3 cm 10 cm Yes…C =  *d. So, C =  (6)  cm cm 2 10 cm Who remembers the relationship between the circumference and the diameter of the circle? cm

Cylinders 3 cm 10 cm Right…length * width. So, * 10 = cm cm 2 10 cm Now that we have the length and width of the rectangular piece, how do we find its area? cm cm 2

Cylinders 3 cm 10 cm Yes…add up all the areas of the pieces. So, = cm cm 2 10 cm Now we have the area of each piece that makes up the surface of the cylinder. What is its surface area then? cm cm 2

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

For those that finish early 1. Often at carnivals or parties, you’ll see a game where you have to guess how many jelly beans are in a jar. Instead of just guessing randomly, suppose you counted 34 jelly beans that were on the bottom of the jar and you saw that there were about 22 jelly beans going up the height. Using our idea of volume, how could you come up with a fairly good estimate on the number of jelly beans in the jar?

Big Ideas from Today’s Lesson The surface area of a cylinder is determined by finding the area of the two circles (top and bottom) and the area of the rectangular piece that connects the two circles. The formula for the surface area of the cylinder is: SA = 2  r  rh Area of 2 circlesArea of rectangular middle

Big Ideas from Today’s Lesson The volume of a cylinder is found the same way as a prism. We find out how many unit cubes could fit inside the bottom layer and then multiply by the number of layers. The formula for the volume of the cylinder is: V =  r 2 * height # of unit cubes that can fit in bottom layer # of identical layers

Homework Pgs. 565 – 567 (5, 7, 9, 16, 22, 29) Pgs. 576 – 577 (5, 13, 14, 21, 22)