3 2 Title
West Texas A&M University - Home
Virtual Math Lab

Beginning Algebra
Tutorial 32: Formulas


 

deskLearning Objectives


After completing this tutorial, you should be able to:
  1. Solve a formula for a given variable.
  2. Solve problems involving formulas.




desk Introduction



In this tutorial we will be solving problems using formulas to help us.   We will be looking at such formulas as area of a rectangle, volume of a sphere, Pythagorean theorem and so on.  After going through this tutorial, you will be an old pro at solving problems involving formulas.

 

 

desk Tutorial



 

Formulas

 
A formula is an equation that involves two or more variables that have a specific relationship with each other.

 
 
Formulas for Some
2-Dimensional Figures


 

Area of a Parallelogram

parallel lines

area of a paralellogram


 

In other words, to get the area of a parallelogram, you multiply the base and height.

Keep in mind that a rectangle and square are two special types of parallelograms, and would follow this same formula. 

So what would be the area of the following parallelogram be? 

parallel lines


 
area of a paralellogram

*base = 15 and height = 9
*multiply

 
The area is 135 square inches.


 

Area of a Triangle

triangle

area of a triangle


 

In other words, to get the area of a triangle, you take one half of the base times the height

So what would be the area of the following triangle?

triangle


 
area of a triangle

*base = 11 and height = 5
*multiply

 
The area is 27.5 square units.


 

Area and Circumference of a Circle

circle

Area:
area of a circle

Circumference:
circle


 

In other words, to get the area of a circle, you take pi times the radius squared.  And to get the circumference of a circle, you take 2 times pi times the radius.

So what would be the area and circumference of the following circle?

circle


 
area of a circle
*Area
*radius = 8
*8 squared is 64
 

*Circumference
*radius = 8
*multiply
 


 
The area is 64 pi square centimeters.

The circumference is 16 pi centimeters.

 
 
Formulas for Some
3-Dimensional Figures


 

Surface Area and Volume of a Rectangular Solid

rectangle

Surface Area:
surface area rectangle solid
 

Volume:
volume of rectangular solid


 

In other words, to get the surface area of a rectangular solid, you take two times the length times the width plus two times the length times the height plus tow times the width times the height. And to get the volume of a rectangular solid, you take the length times the width times the height.

So what would be the surface area and volume of the following rectangular solid?

rectangle


 
surface area rectangle solid
*Surface Area
*length = 5, width = 2 and height = 3 
*multiply
 
 

*Volume
*length = 5, width = 2 and height = 3 
*multiply


 
The surface area  is 62 square feet.

The volume is 30 cubic feet.


 

Surface Area and Volume of a Sphere

sphere

Surface Area:
surface area sphere
 

Volume:
volume shpere


 

In other words, to get the surface area of a sphere, you take four times pi times the radius squared.  And to get the volume of a sphere, you take the 4/3 of pi times the radius cubed.

So what would be the surface area and volume of the following sphere?

sphere


 
surface area sphere
*Surface Area
*radius = 18
*multiply
 
 

*Volume
*radius = 18
*multiply

 


 
The surface area  is 1296 pi square units.

The volume is 7776 pi cubic units.


 

Surface Area and Volume of a Right Circular Cylinder

cylinder

Surface Area:
surface area cylinder
 

Volume:
volume cylinder


 

In other words, to get the surface area of a right circular cylinder, you take two times pi times the radius times the height and add that to two times pi times the radius squared.  And to get the volume of a right circular cylinder, you take pi times the radius squared times the height.

So what would be the surface area and volume of the following right circular cylinder?

cylinder


 
surface area cylinder
*Surface Area
*radius = 5 and height = 10
*multiply
 
 

*Volume
*radius = 5 and height = 10
*multiply


 
The surface area  is 150 pi square millimeters.

The volume is 250 pi cubic millimeters.


 
 
 
Solving a Formula 
for a Specified Variable

 
Basically, you want to get the variable you are solving for alone on one side and everything else on the other side (including variables you are not solving for) using INVERSE operations.
 

Even though there is more than one variable in a formula, you solve for a specific variable using the exact same steps that you do with an equation in one variable, as shown in Tutorial 14: Solving Linear Equations (Putting it all together).
 

It is really easy to get overwhelmed when there is more than one variable involved.  Sometimes your head feels like it is spinning when you see all of those variables.  Isn’t math suppose to be about numbers?  Well, just remember that a variable represents a number, so if you need to move it to the other side of the equation you use inverse operations, just like you would do with a number.


 
 
notebook Example 1:   Solve the equation example 1a for L.

 
Do you recognize this formula?
This happens to be the formula for the perimeter of a rectangle, where P = perimeter, L = length, and W = width.

In this problem, we need to solve for L.  This means we need to get L on one side and EVERYTHING ELSE on the other side using inverse operations.

Let’s solve this formula for L:


 
example 1b

*Inverse of add 2W is sub. 2W
 

*Inverse of mult. by 2 is div. by 2
 

*Formula solved for L
 


 
 
 
notebook Example 2:   Solve the equation example 2a  for r.

 
Do you recognize this formula?
This happens to be the formula for the circumference of a circle, where C = circumference, example 2b = pi,  and r = radius.

In this problem, we need to solve for r.  This means we need to get r on one side and EVERYTHING ELSE on the other side using inverse operations.

Let’s solve this formula for r:


 
example 2b

 

*Inverse of mult. by 2pi is div. by 2pi

*Formula solved for r


 
 
 
notebook Example 3:   Solve the equation example 3afor y.

 
This is an equation for a line. 

In this problem, we need to solve for y.  This means we need to get y on one side and EVERYTHING ELSE on the other side using inverse operations.

Let’s solve this formula for y:


 
example 3b

*Inverse of add 5x is sub. 5x
 

*Inverse of mult. by 4 is div. by 4

*Formula solved for y
 


 
 
 
 
notebook Example 4:   Solve the equation example 4a for h.

 
Do you recognize this formula?
This happens to be the formula for the volume of a rectangular solid, where V = volume, l = length, w = width, and h = height.

In this problem, we need to solve for h.  This means we need to get h on one side and EVERYTHING ELSE on the other side using inverse operations.

Let’s solve this formula for h:


 
example 4b
*Inverse of mult. by lw is div. by lw
 

*Formula solved for h


 
 
 
Solving Problems Involving 
Formulas

 
Step 1: Identify the type(s) of  figure(s) in the problem. 

 
For example, are you working with a circle, cylinder, square, etc?  Are you working with more than one figure?  These are the questions you need to answer.

 
Step 2:  Identify what formula(s) you need. 

 
For example, are you looking for the perimeter, area , volume, etc. of the figure(s) you identify in step 1?

 
Step 3:  Put the problem together.

 
Sometimes the problem is cut and dry and you just simply plug in to a formula and go.

Sometimes you need to do a little figuring.  You may need to add, subtract, or take a fraction of the formula(s) you came up with in step 2.


 
 
Area of a Rectangle

 
notebook Example 5:   One bag of fertilizer will cover 500 square feet of lawn.  Your rectangular lawn is 70 feet by 50 feet.  How many bags of fertilizer will you need to cover it?

 
Step 1: Identify the type(s) of  figure(s) in the problem. 
AND
Step 2:  Identify what formula(s) you need. 

 
Make sure that you read the question carefully several times. 

Since we are needing to find the area of a rectangle,  we can use this formula: example 5a

The variables in this formula represent the following:

A = Area of a rectangle
L = length
W = width


 
Step 3:  Put the problem together.

 
In this problem, 
A = ? = this is the variable we are looking for
L = 70
W = 50
example 5a

Plugging the values into the formula we get:

example 5b


 
First, find the area of the lawn:

 
example 5c

*Multiply

 
For every 500 square feet, you need 1 bag of fertilizer.  So, we need to see how many times 500 sq. feet goes into 3500 sq. feet to find the number of bags of fertilizer needed.

 
example 5d
*Divide

 
FINAL ANSWER: 

7 bags of fertilizer.


 
 
Volume of a Sphere

 
 
notebook Example 6:   The diameter of a beach ball was found to be 18 inches.  What is the volume of this beach ball?

 
Step 1: Identify the type(s) of  figure(s) in the problem. 
AND
Step 2:  Identify what formula(s) you need. 

 
Make sure that you read the question carefully several times. 

Since we are looking for the volume of a sphere,  we can use this formula:

example 6a

The variables in this formula represent the following:

V = volume of a sphere
r = radius


 
Step 3:  Put the problem together.

 
In this problem, 
V = ? = this is the variable we are looking for
r = 9   (radius is half the diameter, so r = 18/2 = 9)

example 6a

Plugging the values into the formula we get:

example 6b


 
example 6c
*Cube 9

*Multiply


 
FINAL ANSWER: 

The volume of the beach ball is example 6dcubic inches.


 
 
Pythagorean Theorem

 
 
notebook Example 7:   A ramp 13 feet long is leaning against a raised platform which is 5 feet above the ground.  What is the distance from the ramp’s contact point with the ground and the base of the platform?

example 7c


 
Step 1: Identify the type(s) of  figure(s) in the problem. 
AND
Step 2:  Identify what formula(s) you need. 

 
Make sure that you read the question carefully several times. 

Since we are looking for the side of a right triangle,  we can use the Pythagorean formula:

example 7a
example 7d

The variables in this formula represent the following:

a and b = legs of the right triangle
c = hypotenuse of the right triangle


 
Step 3:  Put the problem together.

 
In this problem, 
a = ? = this is the variable we are looking for
b = 5
c = 13 

example 7a

Plugging the values into the formula we get:

example 7e


 
example 7b

*Square 5 and 13

*Subtract 25 from both sides
*What squared gives you 144?


 
FINAL ANSWER: 

The distance from the ramp’s contact point with the ground and the base of the platform is 12 feet.


 
 
Add if You are 
Putting Figures Together

 
 
notebook Example 8:   A cylindrical pedestal for a statue is to have a height of 5 feet and a diameter of 2 feet.  The pedestal’s base is to be a rectangular solid that is 9 feet long, 4 feet wide, and 2 feet thick.  What volume of cement is needed to construct the pedestal and its base? 

 
Step 1: Identify the type(s) of  figure(s) in the problem. 
AND
Step 2:  Identify what formula(s) you need. 

 
Make sure that you read the question carefully several times. 

Since part of the problem is looking for the volume of a cylinder, we can use the formula :

example 8d

The variables in this formula represent the following:

example 8g= volume of the cylinder
r = radius
h = height
 

Since part of the problem is looking for the volume of a rectangular solid, we can use also use the formula:

example 8e

The variables in this formula represent the following:

example 8h= Volume of the rectangular solid
l = length
w = width
h = height


 
Step 3:  Put the problem together.

 
In this problem, 
V = ? = this is the variable we are looking for
r = 1   (radius is half the diameter, so r = 2/2 = 1)
h (of cylinder) = 5

l = 9
w = 4
h (of rectangular solid)= 2

If we take the volume of the cylinder and add it to the volume of the rectangular solid, then we will have the volume that we are looking for:

example 8a

Plugging the values into the formula we get:

example 8f


 
example 8b

*Multiply

 
FINAL ANSWER: 

The volume of the beach ball is example 8c cubic inches.


 
 
Subtract if You are 
Taking Out Parts of a Figure 

 
 
notebook Example 9:   Using the figure shown, find the area in square feet of the middle region in the square?

example 9a


 
Step 1: Identify the type(s) of  figure(s) in the problem. 
AND
Step 2:  Identify what formula(s) you need. 

 
Make sure that you read the question carefully several times. 

Since part of the problem involves the area of a square, we can use the formula :

example 9e

The variables in this formula represent the following:

example 9g= area of square
s = side
 

Since part of the problem involves the area of a circle, we can use also use the formula:

example 9f

The variables in this formula represent the following:

example 9h= area of the four quarter circle corners (four quarters = 1 whole circle)
r = radius



 
Step 3:  Put the problem together.

 
In this problem, 
A = ? = this is the variable we are looking for
s = 20
r = 10 

If we take the area of the square and subtract out the area of the four quarter circles (whole circle) we will have the area of the middle region of the given figure above:

example 9b

Plugging the values into the formula we get:

example 9i


 
example 9c

*Square 20 and 10

 
FINAL ANSWER: 

The area of the middle region  is example 9dsquare feet.


 
 
Take a Fraction of a Formula if You 
Only Have a Portion of a Figure 

 
 
notebook Example 10:   A dome is hemispherical in shape with a radius of 16 meters and is built using 8 equal sections.  What formula would describe the surface area of each section?

 
Step 1: Identify the type(s) of  figure(s) in the problem. 
AND
Step 2:  Identify what formula(s) you need. 

 
Make sure that you read the question carefully several times. 

Since we are looking for the surface area of 1/8 of a hemisphere (half of a sphere), we can use the formula :

example 10a

The variables in this formula represent the following:

SA = surface area
r = radius



 
Step 3:  Put the problem together.

 
In this problem, 
SA = ? = this is the variable we are looking for
r = 16 

example 10a

Plugging 16 in for r we get:

example 10d

Simplifying the expression we get: 


 
example 10b

*Multiply

 
FINAL ANSWER: 

The surface area of one section is example 10csquare meters.


 

 

desk Practice Problems


These are practice problems to help bring you to the next level.  It will allow you to check and see if you have an understanding of these types of problems. Math works just like anything else, if you want to get good at it, then you need to practice it.  Even the best athletes and musicians had help along the way and lots of practice, practice, practice, to get good at their sport or instrument.  In fact there is no such thing as too much practice.

To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that  problem.  At the link you will find the answer as well as any steps that went into finding that answer.

 

pencil Practice Problems 1a - 1b: Solve each equation for the specified variable.

 

1a. problem 1a;   for T
(answer/discussion to 1a)
1b. problem 1b;   for y
(answer/discussion to 1b)

 

pencilPractice Problems 2a - 2c: Solve the following word problems.

 

2a.  Sally is training for the Olympics.  She likes to run around  a circular track that has a diameter of 60 yards, 20 times during a workout.  How many yards does she run during her workout?
(answer/discussion to 2a)

 
2b.  A ramp 5 feet long is leaning against a raised platform which is 4 feet above the ground.  What is the distance from the ramp’s contact point with the ground and the base of the platform?
(answer/discussion to 2b)

 
2c. In the figure, ABCD is a square, with each side of length 8 inches.  The width of the border (shaded portion) between the outer square EFGH and ABCD is 2 inches.  Find the area of the border.

problme 2a



 

 

 

desk Need Extra Help on these Topics?


 

The following are webpages that can assist you in the topics that were covered on this page: 
 

http://www.purplemath.com/modules/solvelit.htm
This webpage helps you with solving formulas for a specified variable.

http://www.purplemath.com/modules/perimetr.htm
This webpage involves a combination of problem solving ideas which includes problem solving using formulas.


 

Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.


 

 

 

buffalo top

 


Last revised on August 3, 2011 by Kim Seward.
All contents copyright (C) 2001 - 2010, WTAMU and Kim Seward. All rights reserved.