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    what measures of a cylinder can be determined when given the volume and the height? describe how to find the measure.

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    Volume of a Cylinder

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    Volume of a Cylinder

    A cylinder is a solid composed of two congruent circles in parallel planes, their interiors, and all the line segments parallel to the segment containing the centers of both circles with endpoints on the circular regions.

    The volume of a 3 3

    -dimensional solid is the amount of space it occupies.  Volume is measured in cubic units (

    in 3 , ft 3 , cm 3 , m 3 in3,ft3,cm3,m3

    , et cetera).  Be sure that all of the measurements are in the same unit before computing the volume.

    The volume V V

    of a cylinder with radius

    r r

    is the area of the base

    B B times the height h h . V=Bh  or  V=π r 2 h V=Bh  or  V=πr2h

    Example:

    Find the volume of the cylinder shown. Round to the neatest cubic centimeter.

    Solution

    The formula for the volume of a cylinder is

    V=Bh  or  V=π r 2 h V=Bh  or  V=πr2h .

    The radius of the cylinder is

    8 8

    cm and the height is

    15 15 cm. Substitute 8 8 for r r and 15 15 for h h in the formula V=π r 2 h V=πr2h . V=π (8) 2 (15) V=π(8)2(15) Simplify. V=π(64)(15) ≈3016 V=π(64)(15)≈3016

    Therefore, the volume of the cylinder is about

    3016 3016 cubic centimeters.

    Source : www.varsitytutors.com

    Heights of Cylinders Given Volume

    Discover how to calculate the height of a cylinder when provided with the volume and radius or diameter.

    Back To Finding Missing Dimensions of CylindersBack

    8.2

    Heights of Cylinders Given Volume

    Last Modified: Aug 18, 2016

    [Figure 1]

    Gregory's family just bought a hot tub for their lake house.  The hot tub company told his family that the tub holds 125 cubic feet of water.  Gregory is interested to know how deep the hot tub is.  He measures the diameter of the top and finds that the hot tub is 6 feet across.  What is the height, or depth, of the hot tub?

    In this concept, you will learn how to calculate the height of a cylinder when given the volume and radius or diameter.

    Finding the Height of a Cylinder Given Volume

    Sometimes you will know the volume and radius of a cylinder and you won’t know the height of it. Think about a water tower that is cylindrical in shape. You might know how much volume the tank will hold and the radius of the tank, but not the height of it. When this happens, you can use the formula for the volume of a cylinder to find the missing height:

    V V V V V =π r 2 h =π(2 ) 2 (10) =π(4)(10) =40π =125.6 in . 3

    V=πr2hV=π(2)2(10)V=π(4)(10)V=40πV=125.6 in.3

    [Figure 2]

    Let's look at an example.

    A cylinder with a radius of 2 inches has a volume of 125.6 cubic inches. What is the height of the cylinder?

    The volume and the radius are given, so substitute these into the formula and then solve for

    h h , the height. V 125.6 125.6 125.6 125.6÷ 12.56 10 in =π r 2 h =(3.14)( 2 2 )h =(3.14)(4)h =12.56h =12.56h÷ 12.56 =h

    V=πr2h125.6=(3.14)(22)h125.6=(3.14)(4)h125.6=12.56h125.6÷ 12.56=12.56h÷ 12.5610 in=h

    The height of the cylinder is 10 inches.

    Check your work by substituting the answer in for the height. You should get a volume of 125.6 cubic inches.

    V V V V V =π r 2 h =π(2 ) 2 (10) =π(4)(10) =40π =125.6 in . 3

    V=πr2hV=π(2)2(10)V=π(4)(10)V=40πV=125.6 in.3

    What is the height of a cylinder that has a radius of 6 cm and a volume of 904.32 cubic cm?

    Again, you have been given the volume and the radius. Put this information into the formula along with the value of pi and solve for

    h h , the height. V 904.32 904.32 904.32 904.32÷113.04 8 cm =π r 2 h =(3.14)( 6 2 )h =(3.14)(36)h =113.04 h =113.04h÷ 113.04 =h

    V=πr2h904.32=(3.14)(62)h904.32=(3.14)(36)h904.32=113.04 h904.32÷113.04=113.04h÷ 113.048 cm=h

    The height of this cylinder is 8 centimeters.

    Examples

    Example 1

    Earlier, you were given a problem about Gregory and his family's hot tub.

    To figure this out, use the formula for volume of a cylinder. He already knows the volume of the tub is 125 cubic feet and the diameter is 6 feet.

    First, divide the diameter by 2 and plug the values for volume, pi, and radius into the formula for volume of a cylinder.

    r r =6÷ 2 =3 r =6÷ 2r =3 V 125 =π r 2 h =(3.14)( 3 2 )h

    V=πr2h125=(3.14)(32)h

    Next, square the radius and multiply the values together.

    125 125 =(3.14)( 3 2 )h =(3.14)(9)h

    125=(3.14)(32)h125=(3.14)(9)h

    125 =28.26h 125=28.26h

    Last, divide both sides by 200.96 for the answer, remembering to include the appropriate unit of measurement.

    125 125÷ 28.26 4.42 ft =28.26h =28.26h÷ 28.26 =h

    125=28.26h125÷ 28.26=28.26h÷ 28.264.42 ft=h

    The answer is Gregory's hot tub is 4.42 feet deep.

    Example 2

    Javier wants to construct a cylindrical container to hold enough water for his pet fish. He read that the fish needs to live in 2,110.08 cubic inches of water. If he constructs a tank with a diameter of 16 inches, how tall must he make it so that it holds the right amount of water?

    First, divide the diameter by 2 and plug the values for volume, pi, and radius into the formula for volume of a cylinder.

    r r =16÷ 2 =8 r =16÷ 2r =8 V 2,110.08 =π r 2 h =(3.14)( 8 2 )h

    V=πr2h2,110.08=(3.14)(82)h

    Next, square the radius and multiply the values together.

    2,110.08 2,110.08 =(3.14)( 8 2 )h =(3.14)(64)h

    2,110.08=(3.14)(82)h2,110.08=(3.14)(64)h

    2,110.08 =200.96h 2,110.08=200.96h

    Then, divide both sides by 200.96 for the answer, remembering to include the appropriate unit of measurement.

    2,110.08 2,110.08÷ 200.96 10.5 in =200.96h

    Source : flexbooks.ck12.org

    Volume of a Cylinder

    The volume of a cylinder of radius r and height h is V = πr^2h. Learn the formulas for the volumes of different types of cylinders along with a few solved examples.

    Volume of Cylinder

    The volume of a cylinder is the capacity of the cylinder which calculates the amount of material quantity it can hold. In geometry, there is a specific volume of a cylinder formula that is used to measure how much amount of any quantity whether liquid or solid can be immersed in it uniformly. A cylinder is a three-dimensional shape with two congruent and parallel identical bases. There are different types of cylinders. They are:

    Right circular cylinder: A cylinder whose bases are circles and each line segment that is a part of the lateral curved surface is perpendicular to the bases.Oblique Cylinder: A cylinder whose sides lean over the base at an angle that is not equal to a right angle.Elliptic Cylinder: A cylinder whose bases are ellipses.Right circular hollow cylinder: A cylinder that consists of two right circular cylinders bounded one inside the other.

    What is the Volume of a Cylinder?

    The volume of a cylinder is the number of unit cubes (cubes of unit length) that can be fit into it. It is the space occupied by the cylinder as the volume of any three-dimensional shape is the space occupied by it. The volume of a cylinder is measured in cubic units such as cm3, m3, in3, etc. Let us see the formula used to calculate the volume of a cylinder.

    Definition of a Cylinder

    A cylinder is a three-dimensional solid shape that consists of two parallel bases linked by a curved surface. These bases are like a circular disk in a shape. The line passing from the center or joining the centers of two circular bases is called the axis of the cylinder.

    Volume of Cylinder Formula

    We know that a cylinder resembles a prism (but note that a cylinder is not a prism as it has a curved side face), we use the same formula of volume of a prism to calculate the volume of a cylinder as well. We know that the area of a prism is calculated using the formula,

    V = A × h, where

    A = area of the base

    h = height

    Now we will apply this formula to calculate the volume of different types of cylinders.

    Volume of a Right Circular Cylinder

    We know that the base of a right circular cylinder is a circle and the area of a circle of radius 'r' is πr2. Thus, the volume (V) of a right circular cylinder, using the above formula, is,

    V = πr2h Here,

    'r' is the radius of the base (circle) of the cylinder

    'h' is the height of the cylinder

    π is a constant whose value is either 22/7 (or) 3.142.

    Thus, the volume of cylinder directly varies with its height and directly varies with the square of its radius. i.e., if the radius of the cylinder becomes double, then its volume becomes four times.

    Volume of an Oblique Cylinder

    The formula to calculate the volume of cylinder (oblique) is the same as that of a right circular cylinder. Thus, the volume (V) of an oblique cylinder whose base radius is 'r' and whose height is 'h' is,

    V = πr2h

    Volume of an Elliptic Cylinder

    We know that an ellipse has two radii. Also, we know that the area of an ellipse whose radii are 'a' and 'b' is πab. Thus, the volume of an elliptic cylinder is,

    V = πabh Here,

    'a' and 'b' are the radii of the base (ellipse) of the cylinder.

    'h' is the height of the cylinder.

    π is a constant whose value is either 22/7 (or) 3.142.

    Volume of a Right Circular Hollow Cylinder

    As a right circular cylinder is a cylinder that consists of two right circular cylinders bounded one inside the other, its volume is obtained by subtracting the volume of the inside cylinder from that of the outside cylinder. Thus, the volume (V) of a right circular hollow cylinder is,

    V = π(R2 - r2)h Here,

    'R' is the base radius of the outside cylinder.

    'r' is the base radius of the inside cylinder.

    'h' is the height of the cylinder.

    π is a constant whose value is either 22/7 (or) 3.142.

    How To Calculate the Volume of Cylinder?

    Here are the steps to calculate the volume of cylinder:

    Identify the radius to be 'r' and height to be 'h' and make sure that they both are of the same units.

    Substitute the values in the volume formula V = πr2h.

    Write the units as cubic units.

    Example: Find the volume of a right circular cylinder of radius 50 cm and height 1 meter. Use π = 3.142.Solution:

    The radius of the cylinder is, r = 50 cm.

    Its height is, h = 1 meter = 100 cm.

    Its volume is, V = πr2h = (3.142)(50)2(100) = 785,500 cm3.

    Note: We need to use the formula to find the volume of a cylinder depending on its type as we discussed in the previous section. Also, assume that a cylinder is a right circular cylinder if there is no type given and apply the volume formula to be V = πr2h.

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