Finally, a sphere can be sliced by an infinite number of different planes. Some planes include point , the center of the sphere. Other points do not include the center.
Surface Area of a Sphere
You can infer the formula for the surface area of a sphere by taking measurements of spheres and cylinders. Here we show a sphere with a radius of and a right cylinder with both a radius and a height of and express the area in terms of .
Now try a larger pair, expressing the surface area in decimal form.
Look at a third pair.
Is it a coincidence that a sphere and a cylinder whose radius and height are equal to the radius of the sphere have the exact same surface area? Not at all! In fact, the ancient Greeks used a method that showed that the following formula can be used to find the surface area of any sphere (or any cylinder in which ).
The Surface Area of a Sphere is given by:
Example 1
Find the surface area of a sphere with a radius of 14 feet.
Use the formula.
Example 2
Find the surface area of the following figure in terms of .
The figure is made of one half sphere or hemisphere, and one cylinder without its top.
Now find the area of the cylinder without its top.
Thus, the total surface area is
Volume of a Sphere
A sphere can be thought of as a regular polyhedron with an infinite number of congruent polygon faces. A series polyhedra with an increasing number of faces is shown.
As , the number of faces increases to an infinite number, the figure approaches becoming a sphere.
So a sphere can be thought of as a polyhedron with an infinite number faces. Each of those faces is the base of a pyramid whose vertex is located at , the center of the sphere. This is shown below.
Each of the pyramids that make up the sphere would be congruent to the pyramid shown. The volume of this pyramid is given by:
To find the volume of the sphere, you simply need to add up the volumes of an infinite number of infinitely small pyramids.
Now, it should be obvious that the sum of all of the bases of the pyramids is simply the surface area of the sphere. Since you know that the surface area of the sphere is , you can substitute this quantity into the equation above.
Finally, as increases and the surface of the figure becomes more “rounded,” , the height of each pyramid becomes equal to , the radius of the sphere. So we can substitute for . This gives:
We can write this as a formula.
Volume of a Sphere
Given a sphere that has radius
Example 3
Find the volume of a sphere with radius .
Use the postulate above.
Example 4
A sphere has a volume of . Find its diameter.
Use the postulate above. Convert to an improper fraction,
Since , the diameter is .
Review Questions
Find the radius of the sphere that has a volume of .
Determine the surface area and volume of the following shapes:
The radius of a sphere is . Find its volume and total surface area.
A sphere has a radius of . A right cylinder, having the same radius has the same volume. Find the height and total surface area of the cylinder.
In problems 6 and 7 find the missing dimension.
Tennis balls with a diameter of are sold in cans of three. The can is a cylinder. Assume the balls touch the can on the sides, top and bottom. What is the volume of the space not occupied by the tennis balls?
A sphere has surface area of . Find its volume.
A giant scoop, operated by a crane, is in the shape of a hemisphere of . The scoop is filled with melted hot steel. When the steel is poured into a cylindrical storage tank that has a radius of , the melted steel will rise to a height of how many inches?
Review Answers
Height of molten steel in cylinder will be
Similar Solids
Learning Objectives
Find the volumes of solids with bases of equal areas.
Introduction
You’ve learned formulas for calculating the volume of different types of solids—prisms, pyramids, cylinders, and spheres. In most cases, the formulas provided had special conditions. For example, the formula for the volume of a cylinder was specific for a right cylinder.
Now the question arises: What happens when you consider the volume of two cylinders that have an equal base but one cylinder is non-right—that is, oblique. Does an oblique cylinder have the same volume as a right cylinder if the two share bases of the same area?
Parts of a Solid
Given, two cylinders with the same height and radius. One cylinder is a right cylinder, the other is oblique. To see if the volume of the oblique cylinder is equivalent to the volume of the right cylinder, first observe the two solids.
Since they both have the same circular radius, they both have congruent bases with area:
Now cut the right cylinder into a series of cross-section disks each with height and radius .
It should be clear from the diagram that the total volume of the disks is equal to the volume of the original cylinder.
Now start with the same set of disks. Shift each disk over to the right. The volume of the shifted disks must be exactly the same as the unshifted disks, since both figures are made out of the same disks.
It follows that the volume of the oblique figure is equal to the volume of the original right cylinder.
In other words, if the radius and height of each figure are congruent:
The principle shown above was developed in the seventeenth century by Italian mathematician Francisco Cavalieri. It is known as Cavalieri’s Principle. (Liu Hui also discovered the same principle in third-century China, but was not given credit for it until recently.) The principle is valid for any solid studied in this chapter.
Volume of a Solid Postulate (Cavalieri’s Principle):
The volumes of two objects are equal if the areas of their corresponding cross-sections are in all cases equal. Two cross-sections correspond if they are intersections of the figure with planes equidistant from a chosen base plane.
Example 1
Prove (informally) that the two circular cones with the same radius and height are equal in volume.
As before, we can break down the right circular cone into disks.
Now shift the disks over.
You can see that the shifted-over figure, since it uses the very same disks as the straight figure, must have the same volume. In fact, you can shift the disks any way you like. Since you are always using the same set of disks, the volume is the same.
Keep in mind that Cavalieri’s Principle will work for any two solids as long as their bases are equal in area (not necessarily congruent) and their cross sections change in the same way.
Example 2
A rectangular pyramid and a circular cone have the same height, and base area. Are their volumes congruent?
Yes. Even though the two figures are different, both can be computed by using the following formula:
Since
Then
Similar or Not Similar
Two solids of the same type with equal ratios of corresponding linear measures (such as heights or radii) are called similar solids.
To be similar, figures need to have corresponding linear measures that are in proportion to one another. If these linear measures are not in proportion, the figures are not similar.
Example 1
Are these two figures similar?
If the figures are similar, all ratios for corresponding measures must be the same.
The ratios are:
Since the three ratios are equal, you can conclude that the figures are similar.
Example 2
Cone A has height and radius . Cone B has height and radius . Are the two cones similar?
If the figures are similar all ratios for corresponding measures
must be the same.
The ratios are:
Since the ratios are different, the two figures are not similar.
Compare Surface Areas and Volumes of Similar Figures
When you compare similar two-dimensional figures, area changes as a function of the square of the ratio of
For example, take a look at the areas of these two similar figures.
The ratio between corresponding sides is:
The ratio between the areas of the two figures is the square of the ratio of the linear measurement:
This relationship holds for solid figures as well. The ratio of the areas of two similar figures is equal to the square of the ratio between the corresponding linear sides.
Example 3
Find the ratio of the surface area between the two similar figures C and D.
Since the two figures are similar, you can use the ratio between any two corresponding measurements to find the answer. Here, only the radius has been supplied, so:
The ratio between the areas of the two figures is the square of the ratio of the linear measurements:
Example 4
If the surface area of the small cylinder in the problem above is 80π, what is the surface area of the larger cylinder?
From above we, know that:
So the surface area can be found by setting up equal ratios
Solve for .
The ratio of the volumes of two similar figures is equal to the cube of the ratio between the corresponding linear sides.
Example 5
Find the ratio of the volume between the two similar figures C and D.
As with surface area, since the two figures are similar you can use the height, depth, or width of the figures to find the linear ratio. In this example we will use the widths of the two figures.
The ratio between the volumes of the two figures is the cube of the ratio of the linear measurements:
Does this cube relationship agree with the actual measurements? Compute the volume of each figure.
As you can see, the ratio holds. We can summarize the information in this lesson in the following postulate.
Similar Solids Postulate:
If two solid figures, A and B are similar and the ratio of their linear measurements is , then the ratio of their surface areas is:
The ratio of their volumes is:
Scale Factors and Models
The ratio of the linear measurements between two similar figures is called the scaling factor. For example, we can find the scaling factor for cylinders and by finding the ratio of any two corresponding measurements.
Using the heights, we find a scaling factor of:
You can use a scaling factor to make a model.
Example 6
Doug is making a model of the Statue of Liberty. The real statue has a height of and a nose that is in length. Doug’s model statue has a height of . How long should the nose on Doug’s model be?
First find the scaling factor.
To find the length of the nose, simply multiply the height of the model’s nose by the scaling factor.
In inches, the quantity would be:
Example 7
An architect makes a scale model of a building shaped like a rectangular prism. The model measures in height, in width, and in depth. The real building will be tall. How wide will the real building be?
First find the scaling factor.
To find the width, simply multiply the width of the model by the scaling factor.
Review Questions
How does the volume of a cube change if the sides of a cube are multiplied by 4? Explain.
In a cone if the radius and height are doubled what happens to the volume? Explain.
In a rectangular solid, is the sides are doubled what happens to the volume? Explain.
Two spheres have radii of and . What is the ratio of their volumes?
The ratio of the volumes of two similar pyramids is . What is the ratio of their total surface areas?
Are all spheres similar?
Are all cylinders similar?
Are all cubes similar? Explain your answers to each of these.
The ratio of the volumes of two tetrahedron is . The smaller tetrahedron has a side of length . What is the side length of the larger tetrahedron?
Refer to these two similar cylinders in problems 8 – 10:
What is the similarity ratio of cylinder A to cylinder B?
What is the ratio of surface area of cylinder A to cylinder B?
What is the ratio of the volume of cylinder B to cylinder A?
Review Answers
The volume will be greater.
Volume will be greater.
The volume will be greater (volume of first rectangular solid)
All spheres and all cubes are similar since each has only one linear measure. All cylinders are not similar. They can only be similar if the ratio of the radii the ratio of the heights.
Chapter 12: Transformations
Translations
Learning Objectives
Graph a translation in a coordinate plane.
Recognize that a translation is an isometry.
Use vectors to represent a translation.
Introduction
Translations are familiar to you from earlier lessons. In this lesson, we restate our earlier learning in terms of motions in a coordinate plane. We’ll use coordinates and vectors to express the results of translations.
Translations
Remember that a translation moves every point a given horizontal distance and/or a given vertical distance. For example, if a translation moves the point to the right and up, to then this translation moves every point the same way.
The original point (or figure) is called the preimage, in this case . The translated point (or figure) is called the image, in this case , and is designated with the prime symbol.
Example 1
The point in a translation becomes the point . What is the image of in the same translation?
Point moved to the left and down to reach . will also move to the left and down.
is the image of .
Notice the following:
Since the endpoints of and moved the same distance horizontally and vertically, both segments have the same length.
Translation is an Isometry
An isometry is a transformation in which distance is “preserved.” This means that the distance between any two points is the same as the distance between the images of the points.
Did you notice this in example 1 above?
(since they are both equal to )
Would we get the same result for any other point in this translation? Yes. It’s clear that for any point , the distance from to will be . Every point moves to its image.
This is true in general.
Translation Isometry Theorem
Every translation in the coordinate plane is an isometry.
You will prove this theorem in the Lesson Exercises.
Vectors
Let’s look at the translation in example 1 in a slightly different way.
Example 2
The point in a translation is the point . What is the image of in the same translation?
The arrow from to is called a vector, because it has a length and a direction. The horizontal and vertical components of the vector are and respectively.
To find the image of , we can apply the same transformation vector to point . The arrowhead of the vector is at .
The vector in example 2 is often represented with a boldface single letter .
The horizontal component of vector is .
The vertical component of vector is .
The vector can also be represented as a number pair made up of the horizontal and vertical components.
The vector for this transformation is
Example 3
A triangle has vertices , and . The vector for a translation is . What are the vertices of the image of the triangle?
Add the horiz
ontal and vertical components to the and coordinates of the vertices.
Challenge: Can you describe what this transformation does to the original triangle?
Further Reading
Vectors are used in physics to represent forces, velocity, and other quantities. Learn more about vectors at: http://en.wikipedia.org/wiki/Vector_(spatial)
Lesson Summary
You can think of a translation as a way to move points in a coordinate plane. And you can be sure that the shape and size of a figure stays the same in a translation. For that reason a translation is called an isometry. (Note: Isometry is a compound word with two roots in Greek, “iso” and “metry.” You may know other words with these same roots, in addition to “isosceles” and “geometry.”)
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