unit 3 • circles and volume lesson 5: explaining and...

UNIT 3 • CIRCLES AND VOLUMELesson 5: Explaining and Applying Area and Volume Formulas

Instruction

CCGPS Analytic Geometry Teacher Resource © Walch EducationU3-225

IntroductionThink about the dissection arguments used to develop the area of a circle formula. These same arguments can be used to develop the volume formulas for cylinders, pyramids, and cones. You have already used volume formulas for prisms, cylinders, pyramids, and cones, but how are the formulas derived? In this lesson, you will learn how to find the volumes of cylinders, pyramids, and cones. As part of learning the formulas, you will see proofs of why the formulas work for those objects. The real world is filled with these objects. Using the formulas of volume for these objects expands your problem-solving skills.

Key Concepts

• The formula for finding the volume of a prism is V = length • width • height. This can also be shown as V = area of base • height.

• Remember to use cubic units or volume measures when calculating volume. Some examples are cubic feet (ft3), cubic meters (m3), liters (L), and gallons (gal).

Cylinders

• A cylinder has two bases that are parallel. This is also true of a prism.

• Bonaventura Cavalieri, an Italian mathematician, formulated Cavalieri’s Principle. This principle states that the volumes of two objects are equal if the areas of their corresponding cross sections are in all cases equal.

• This principle is illustrated by the diagram below. A rectangular prism has been sliced into six pieces and is shown in three different ways.

• The six pieces maintain their same volume regardless of how they are moved.

Prerequisite Skills

This lesson requires the use of the following skills:

• understanding and using formulas for the volume of prisms, cylinders, pyramids, and cones

• understanding and applying the formula for circumference


Instruction

CCGPS Analytic Geometry Teacher Resource U3-226

© Walch Education

• Cavalieri’s Principle describes how each piece is a thin slice in the plane of the prism.

• If each thin slice in each object has the same area, then the volumes of the objects are the same.

• The following diagram shows a prism, a prism at an oblique angle, and a cylinder.

hBBB

• The three objects meet the two criteria of Cavalieri’s Principle. First, the objects have the same height. Secondly, the areas of the objects are the same when a plane slices them at corresponding heights. Therefore, the three objects have the same volume.

• A square prism that has side lengths of r π will have a base area of 2π π π• =r r r on every plane that cuts through it. The same is true of a cylinder, which has a radius, r.

• The base area of the cylinder will be π r 2. This shows how a square prism and a cylinder can have the same areas at each plane.

• Another way of thinking about the relationship of a polygonal prism and the cylinder is to remember the earlier proof about the area of a circle.

• A cylinder can be thought of as a prism with an infinite number of sides.

• The diagram below shows a polygonal prism with 200 sides.


Instruction


• The area of its base can be calculated by dividing it into quadrilaterals and triangles, but its base is approaching the limit of being a circle.

• If the base area and the height of a prism and a cylinder are the same, the prism and cylinder will have the same volume.

• The formula for finding the volume of a cylinder is V = π r 2 • h.

Pyramids

• A pyramid is a solid or hollow polyhedron object that has three or more triangular faces that converge at a single vertex at the top; the base may be any polygon.

• A polyhedron is a three-dimensional object that has faces made of polygons.

• A triangular prism can be cut into three equal triangular pyramids.

Triangular prisms

Corresponding triangular pyramids


Instruction


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• A cube can be cut into three equal square pyramids.

• This dissection proves that the volume of a pyramid is one-third the volume of a prism:

V B hpyramid =1

3• .

Cones

• A cone is a solid or hollow object that tapers from a circular base to a point.

• A cone and a pyramid use the same formula for finding volume.

• This can be seen by increasing the number of sides of a pyramid.

• The limit approaches that of being a cone.

• A pyramid with 100 sides follows. With such a large number of sides, it looks like a cone.

• The formula for the volume of a cone is V r hcone =1

32π • .

• Cavalieri’s Principle shows how pyramids and cones have the same volume.

• The diagram that follows shows cross sections of areas with the same planes.

• Each object has the same area at each cross section.


Instruction


• Therefore, the volumes of both objects are the same.

Common Errors/Misconceptions

• using the prism volume formula for a pyramid or a cone

• calculating the wrong base area for a pyramid due to not noticing the shape of the pyramid’s base (e.g., triangle, square)

• forgetting to express a calculation in the correct units for volume (e.g., m3, liters)


Instruction


© Walch Education

Example 1

Find the dimensions for a cylinder that has the same volume as a square prism with a base area of 9 square meters. The cylinder and the square prism should both have heights of 5 meters.

5 m5 m

1. Determine the relationship between two objects with the same volume.

Cavalieri’s Principle states that if two objects have the same area in every plane, or cross section, then their volumes are the same. The cylinder and prism have uniform width and length throughout their heights. Both need to have the same height. Therefore, in order for the cylinder and prism to have the same volume, they need to have the same areas for their bases.

2. Set up the formulas for the area of the base of the cylinder and the area of the base of the prism so that they are equal.

The formula for the area of the circular base of the cylinder is A = π r 2, where r is the radius of the base of the cylinder.

The area of the base of the prism is 9 square meters.

Abase of square prism

= Abase of cylinder

Set the areas of the bases equal.

9 = π r 2 Substitute the known information to form an equation.

Guided Practice 3.5.2


Instruction


3. Solve the equation for r.

9 = π r 2

92

π= r Divide both sides by π.

2.864788976 ≈ r 2

± ≈2 864788976. r Take the square root of both sides.

r ≈ 1.692568751 The length of the radius must be a positive number.

For the cylinder to have the same volume as the prism, the cylinder’s base must have a radius of 1.692568751.

Substitute the value of the radius into the equation to check your work.

9 = π • r 2

9 = π • (1.692568751)2

9 ≈ 9.0000000001

The difference is infinitesimal. The two objects essentially have bases that have the same area, 9 m2.

4. Calculate the volume for each object.

The volume for a prism or a cylinder can be seen as many thin slices or cross sections stacked on top of one another. Thus, their volume formulas are the same: V = B • h, where B is the area of the base of the object.

Prism Cylinder

V = B • h V = B • h

V = 9 • 5 V ≈ 9.0000000001 • 5

V = 45 m3 V ≈ 45 m3


Instruction


© Walch Education

5. Verify that the two objects will have the same area at a height of 1 meter or any other height.

Yes, this is true. At every height, both objects have uniform dimensions that are equal to the dimensions of their bases. Therefore, using Cavalieri’s Principle, both objects, the prism and the cylinder, have the same volume.

Example 2

Dissect a cube to prove that it has three times the volume of a square pyramid that has a base of the same area as the cube.

1. A cube has 6 equal sides. Divide the cube into 6 equal pyramids.

Fix a point in the exact center of the cube.

Place 6 square pyramids in the cube, one on each side of the cube facing inward toward the center of the cube so that the vertex of each pyramid touches the exact center of the cube. Each pyramid will have a height equal to one half the height of the cube. The sides of the pyramids will rest against one another with no gaps or spaces between them. The volume of the cube is completely filled with the 6 pyramids.


Instruction


2. Derive the formula for the volume of a pyramid by using the formula for the volume of the cube. A cube has equal sides, so the height of the cube equals the side length.

Vcube

= B • s, where s equals the length of a side of the cube.

Substitute the pyramid height, h, for the height of the cube, s. The height of one of the pyramids is equal to half the height of the cube. Therefore, the height of the cube equals two of the pyramids, or 2h.

Vcube

= B • 2h

There are 6 pyramids contained in the cube.

Therefore, to find the volume of one pyramid, divide the volume of the cube by 6.

=•

VB h2

6pyramid

V B hpyramid =1

3• Simplify

2

6 to equal

1

3.

3. Multiply the formula for the volume of the pyramid times 3 and compare it with the formula for the volume of the cube.

31

3B h B h• •

=

The volume of the cube, a prism, is equal to B • h. This shows that a cube has three times the volume of a square pyramid (or that a square pyramid has one-third the volume of a cube).


Instruction


© Walch Education

Example 3

Find the dimensions for a cone that has the same volume as a pyramid of the same height as the cone. Both the cone and the pyramid have a height of 2 meters. The volume of the pyramid is 3 cubic meters. A cone and a pyramid both taper to a point or vertex at the top. The “slant” of the taper is linear, meaning it is a straight line. The dimensions of both the cone and the pyramid change at a constant rate from base to tip.

2 m2 m

1. Cavalieri’s Principle states that the pyramid and cone will have the same volume if the area of each cross section of a plane is the same at every height of the two objects. This means that if the cone and pyramid have bases of equal area, then their volumes will also be equal.


Instruction


2. Set up an equation to find the radius of the cone.

The volume of the pyramid is 3 m3.

Both objects must have bases of equal area to have the same volume.

The area of the base of the pyramid can be found by solving for B in the formula for the volume of a pyramid.

Vpyramid

= 1

3B • h

3 = 1

3B • 2 Substitute the volume and the height of the pyramid.

3 = 2

3B Simplify.

3

2 • 3 = B Multiply both sides by

3

2.

B=9

2Simplify.

B = 4.5

The area of the base of the pyramid is 4.5 square meters.

Apyramid base

= Acone base

= Acircle

= π r 2 Set up the equation.

4.5 = π r 2Substitute the value found for the area of the base of the pyramid.

rπ

=4.5 2 Divide both sides by π.

rπ

± =4.5

Take the square root of both sides.

r ≈ 1.19682684

The radius of the cone is approximately 1.197 meters.


Instruction


© Walch Education

Example 4

A jet fuel storage tank near a large airport is a cylinder that has a radius of 12.5 meters and a height of 15.854 meters. How many gallons of jet fuel will the tank hold? There are 264.172 gallons in 1 cubic meter.

1. Find the volume of the cylinder.

V = B • h

V = π r 2 • hSubstitute the expression for the area of a circle.

V = π(12.52) • 15.854 Substitute the known dimensions of the tank.

V ≈ 7782.31 m3

The volume of the jet fuel tank is approximately 7,782.31 cubic meters.

2. Find the number of gallons in 7,782.31 cubic meters.

7782.31m •264.172 gallons

1m7 ga3

3 = 2 055 868 39, , . lllons

The storage tank will hold about 2,055,869 gallons of jet fuel.


Instruction


Example 5

A new art museum is being built in the shape of a square pyramid. The height will be 50 meters. The art museum needs 86,400 cubic meters of space inside. What should be the side lengths of the base of the pyramid?

1. Use the formula for the volume of a pyramid to find the unknown side lengths of the base.

V B h=1

3•

86 4001

350, •= B Substitute the known values.

3 86 400

50

• ,= B Multiply both sides by

3

50.

B = 5184

The base is 5,184 square meters.

2. Find the side length of the base.

The base is a square. It has an area of side • side.

Find the length of one side by taking the square root of the area of the base.

± =5184 72

Take the positive square root since length is a positive number.

The length of each side of the art museum’s pyramid base should be 72 meters.

unit 3 • circles and volume lesson 5: explaining and...

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