Volume

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 Site: TBAISD Moodle Course: Michigan Geometry Preview 2012 Book: Volume Printed by: Guest user Date: Monday, October 22, 2018, 06:25 PM

Definition

The volume of a three-dimensional object is the amount of space inside the object. All three-dimensional objects have basically the same formula for finding their volume:

Volume = Bh
where B is the area of the base and h is its height.

The shape of the base depends on the object. Some 3D shapes have circular bases while others could have bases that are squares, triangles, or other polygons.  If you visualize the shape of the base filling the figure, then this formula should make sense for our 5 geometric solids.   Volumes of these objects will always be given in cubic units - inches3, cm3, etc. because of their three dimensions.

The remainder of this book will discuss how to find the volumes of prisms, cylinders, pyramids, cones and spheres.

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Prisms

As noted previously, the volume of a prism is given by the formula:

V = Bh
where B is the area of the base and h represents the height of the prism.

The prism above is a rectangular prism.  Find the area of the base, which is a rectangle.

B = 6 x 2 = 12 units2

Now, multiply the area of the base by the height of the prism, which is 3:

V = 12 units2 × 3 units  = 36 units3

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Example 1

Find the volume of the triangular prism below.

Step 1.  Find the area of one of the congruent bases.

Step 2.  Multiply the area of the base by the height between the bases.

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Example 2

Find the volume of the regular pentagonal prism below.

Step 1.  Find the area of one of the congruent bases.

A = 0.5 · a · P

A = 0.5 · 8 · 60

A = 240 ft2

Step 2.  Multiply the area of the base by the height between the two congruent bases.

V = B · h

V = 240 ft2 ·  22 ft

V = 5280 ft3

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Video Lesson

For a video lesson on volumes of prisms, select the following link:

Volumes of Prisms

Guided Practice

To solidify your understanding of volume of prisms, visit the following link to Holt, Rinehart, and Winston Homework Help Online. It provides examples, video tutorials and interactive practice with answers available. The Practice and Problem Solving section has two parts. The first part offers practice with a complete video explanation for the type of problem with just a click of the video icon. The second part offers practice with the solution for each problem only a click of the light bulb away.

Guided Practice

Cylinders

The volume of a cylinder is also found by using the formula V = Bh.  The only difference is that the base of the cylinder is always a circle.   Replace B in the formula with the formula for the area of a circle.

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Example

Find the volume of the cylinder below.

Step 1.  Identify the radius and height of the cylinder.

The radius is 7 cm and the height is 12 cm.

Step 2.  Substitute the radius and height into the formula and calculate.

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Video Lessons

For a video lesson on finding the volume of right cylinders, select the following link:

Volume of Cylinders

For a video lesson on finding the volume of oblique cylinders, select the following link:

Volume of Cylinders

Pyramids

Like the volume of a prism, the volume of a pyramid uses the area of its base and its height.  As with prisms, finding the area of the base depends on the shape of the base.  However, the pyramid comes to a point so its volume is going to be less than that of a prism with an equivalent base. The relationship works out that the volume of the pyramid is one-third that of the prism.

For an activity that helps you see why a pyramid's volume is one-third that of a prism with equivalent base, select the following link:

Volume Activity

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Example 1

The following figure is a right pyramid with an isosceles triangle base. Find the volume of the pyramid if the height is 20 cm.

Step 1.  Find the height of the base.

When the height of the triangular base is unknown, use Pythagorean Theorem.

Step 2.  Find the area of the base.

Step 3.  Substitute the area of the base and the height into the volume formula and calculate the volume.

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Example 2

Find the volume of the square pyramid below:

Step 1.  Find the area of the base.

A = 6 · 6

A = 36

Step 2.  Substitute the area of the base and the height into the volume formula and calculate the volume.

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Online Lesson

To review finding the volume of pyramids with different polygon bases, select the following link:

Volume of Pyramids

Video Lesson

For a video lesson on finding the volume of pyramids, select the following link:

Volume of Pyramids

Interactive

For interactive practice problems finding the volume of prisms and cylinders, select the folowing link:

Practice Volume Problems

Cones

Like the volume formula of a pyramid, the volume of a cone is one-third of the volume of a cylinder with an equivalent base.  Recall that the volume of a cylinder is two times pi times the square of the radius times the height.

where r is the radius of the circle and h is the height of the cone.

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Example

Find the volume of the cone below:

Step 1.  Identify the radius and height of the cone.

The radius is 5.7 cm and the height is 12 cm.

Step 2.  Substitute the radius and the height into the volume formula and calculate.

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Video Lesson

For a video lesson on finding the volume of cones, select the following link:

Volume of Cones

Guided Practice

To solidify your understanding of volume of pyramids and cones, visit the following link to Holt, Rinehart, and Winston Homework Help Online. It provides examples, video tutorials and interactive practice with answers available. The Practice and Problem Solving section has two parts. The first part offers practice with a complete video explanation for the type of problem with just a click of the video icon. The second part offers practice with the solution for each problem only a click of the light bulb away.

Guided Practice

Sphere

The volume of a sphere is similar to that of a cylinder.  If you place a sphere with the same length of radius as the base of the cylinder inside a cylinder, it does not fill the cylinder.  Therefore, the volume of a sphere must be smaller than the volume of a cylinder with its base congruent to the great circle of the sphere.  The volume of a sphere is actually two-thirds that of the cylinder.  The video below verifies this fact, please watch it prior to reading the rest of this page.

Now using the information above, let's transform the formula for the volume of a cylinder into the volume of a sphere.  The height of the cylinder is the length of the diameter of the sphere.

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Example

Find the volume of the sphere below.

Step 1.  Find the radius of the sphere.

The radius of the sphere is 9.6 m.

Step 2.  Substitute the radius into the volume formula.

Step 3.  Use a calculator to approximate the volume.

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Video Lesson

For a video lesson on finding the volume of a sphere, select the following link:

Volume of a Sphere

Hemisphere

A hemisphere is half of a sphere so its volume is half of the volume of a sphere.

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Example

Calculate the volume of a hemisphere with a radius of 10 cm.

Step 1.  Identify the radius of the hemisphere.

The radius of the hemisphere is 10 cm.

Step 2.  Substitute the radius into the volume formula.

Step 3.  Use a calculator to approximate the volume.

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Guided Practice

To solidify your understanding of volume of spheres and hemispheres, visit the following link to Holt, Rinehart, and Winston Homework Help Online. It provides examples, video tutorials and interactive practice with answers available. The Practice and Problem Solving section has two parts. The first part offers practice with a complete video explanation for the type of problem with just a click of the video icon. The second part offers practice with the solution for each problem only a click of the light bulb away.

Guided Practice

Interactive

For an interactive comparing volumes of shapes, select the following link:

Comparison of Volumes

Similar Figures

What effect will multiplying the sides of a 3D shape by a factor of 3 have on its volume? Let's see what happens using a rectangular prism.

First find the volume of the original shape. The volume of a prism is found by multiplying the area of the base, in this case a rectangle, by the height.

Volume = (3 × 4) × 5 = 12 × 5 = 60 cm3

If we apply a scale factor of 3 to all of the sides, we need to multiply each dimension by 3. We now have a (3 × 3) × (4 × 3) × (5 × 3)  and its volume is:

Volume of similar figure = 9 × 12 × 15 = 1620 cm3

How much larger is the new volume compared to the original volume? We can find out by taking the new volume and dividing it by the original.

1620 ÷ 60 = 27

The new volume is 27 times larger than the original when we applied a scale factor of 3. Volume is represented by cubic units and 27 is a perfect cube of 3.

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Scale Factor

The previous example allowed you to explore the relationship between the scale factor of a dilation to the volume of its image. If the similarity ratio of two similar figures is r:t, then the ratio of their volumes is r3:t3.

Example  Given the two figures below, if the volume of the first cylinder is 283 mm 3, what is the volume of the second cylinder?

Step 1.  Find the similarity ratio of the cylinders.

Step 2.  Find the similarity ratio for the volume of the cylinders.

Step 3.  Use a proportion to find the volume of the second cylinder.

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Interactive

For an interactive to investigate the effect of side lengths on the volume and surface area of similar figures, select the following link:

Side Lengths, Volume and Surface Area

Online Lesson

For an online lesson on the effects of scale factors on volume, select the following link:

Scale Factors

Video Lessons

For a video lesson on finding volume of similar figures, select the following link:

Finding Volume Using Similar Figures

How does changing the dimensions of a shape change its volume? For a video lesson on the effect of changing the dimensions, select the following link:

Exploring Effects of Changing Dimensions

Applications

There are many applications of volume and surface area including:  maximum-minimum problems, packaging design, and population density.  To explore some of the many lessons about applications of surface area and volume, select the links below:

Online Lesson - Density of Elements

Video Lesson - Density of Water

Video Lesson - Composite Figures

Problem

For an application problem about finding the volume of 3-dimensional figures, select the link below:

Propane Tank Problem

Rubric

To see a rubric for the previous application problem, select the link below:

Rubric for 3-D Application Problem

Practice

Volume Worksheet

Sources

Sources used in this book:

CutOutFoldUp, "The Volume of a Pyramid is One-Third that of a Pyramid." Accessed 6/29/2012. http://www.cutoutfoldup.com/971-the-volume-of-a-pyramid-is-one-third-that-of-a-prism.php.

Embracing Mathematics, Assessment & Technology in High Schools; A Michigan Mathematics & Science Partnership Grant Project

Holt, Rinehart & Winston, "Exploring Effects of Changing Dimensions." http://my.hrw.com/math06_07/nsmedia/lesson_videos/geo/player.html?contentSrc=6819/6819.xml. (accessed 6/29/2012).

Holt, Rinehart & Winston, "Finding Volume Using Similar Figures." http://my.hrw.com/math06_07/nsmedia/lesson_videos/msm2/player.html?contentSrc=8226/8226.xml. (accessed June 29, 2012).

Holt, Rinehart & Winston, "Spatial Reasoning." http://my.hrw.com/math06_07/nsmedia/homework_help/geo/geo_ch10_06_homeworkhelp.html (accessed 6/13/2011).

Holt, Rinehart & Winston, "Spatial Reasoning." http://my.hrw.com/math06_07/nsmedia/homework_help/geo/geo_ch10_07_homeworkhelp.html. (accessed 6/29/2012).

Holt, Rinehart & Winston, "Spatial Reasoning."  http://my.hrw.com/math06_07/nsmedia/homework_help/geo/geo_ch10_08_homeworkhelp.html. (accessed 6/29/2012).

Holt, Rinehart & Winston, "Volume of a Cylinder." http://my.hrw.com/math06_07/nsmedia/lesson_videos/msm1/player.html?contentSrc=6077/6077.xml. (accessed 6/29/2012).

Hotmath.com, "Hotmath Practice Problems." http://hotmath.com/help/gt/genericprealg/section_9_6.html (accessed 6/13/2011).