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Sometimes equation and formula are used interchangeably, but I was wondering if there is a difference.

For example, suppose we can calculate a car's fuel efficiency as:

mpg = distance traveled in miles / the fuel used in a gallon

Is that an equation or formula?

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    $\begingroup$ According to mathsisfun.com/algebra/equation-formula.html, a formula shows a relationship between 2 or more variables, while an equation doesn't necessarily. $\endgroup$
    – endolith
    Aug 11, 2014 at 15:30
  • $\begingroup$ I was told by my guide that a very simple expression (for an unknown in terms of known) is better called an equation rather than a formula. I am noting this down here as I didn't see such a distinction based on the complexity of the expression in any of the answers. $\endgroup$
    – Cyriac Antony
    Jun 28, 2019 at 5:31

14 Answers 14

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An equation is any expression with an equals sign, so your example is by definition an equation. Equations appear frequently in mathematics because mathematicians love to use equal signs.

A formula is a set of instructions for creating a desired result. Non-mathematical examples include such things as chemical formulas (two H and one O make H2O), or the formula for Coca-Cola (which is just a list of ingredients). You can argue that these examples are not equations, in the sense that hydrogen and oxygen are not "equal" to water, yet you can use them to make water.

Mathematicians have long since realized that when it comes to numbers, certain formulas can be expressed most succinctly as equations. For example, the Pythagorean Theorem $a^2+b^2=c^2$ can be thought of as a formula for finding the length of the side of a right triangle, but it turns out that such a length is always equal to a combination of the other two lengths, so we can express the formula as an equation. The key idea is that the equation captures not just the ingredients of the formula, but also the relationship between the different ingredients.

In your case, "mpg = distance/gallons" is best understood as "a formula in the form of an equation", which means that in this instance the two words are interchangeable.

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    $\begingroup$ There is also the technical meaning of "formula" as a well-formed arrangement of lexical pieces of a formal language, which I think is possibly more to the point here. In which case equations are a special case of formula. $\endgroup$
    – Malice Vidrine
    Feb 27, 2014 at 22:21
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    $\begingroup$ Can you give an example of a formula without an equals sign? $\endgroup$
    – ksoo
    Apr 9, 2014 at 13:15
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    $\begingroup$ So tl;dr formula = algorithm? $\endgroup$
    – MarcusJ
    Jan 9, 2017 at 18:15
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An equation is meant to be solved, that is, there are some unknowns. A formula is meant to be evaluated, that is, you replace all variables in it with values and get the value of the formula.

Your example is a formula for mpg. But it can become an equation if mpg and one of the other value is given and the remaining value is sought.

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    $\begingroup$ So, is $y=3x+1$ the "formula for a line in the plane" or an equation for a line in the plane? $\endgroup$
    – Arturo Magidin
    May 10, 2011 at 2:22
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    $\begingroup$ @lhf: Suffice it to say, I don't think I agree with your dichotomy. $\endgroup$
    – Arturo Magidin
    May 10, 2011 at 2:25
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    $\begingroup$ Better to ask this at the "english stack exchange". I think there are really sensical members there... $\endgroup$
    – Gottfried Helms
    May 10, 2011 at 9:24
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    $\begingroup$ @Alexander, so a formula is like a dead equation? :-) $\endgroup$
    – lhf
    May 10, 2011 at 12:02
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    $\begingroup$ @lhf, that is probably the best comment on MSE. $\endgroup$
    – Mariano Suárez-Álvarez
    Feb 25 at 11:16
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I'd say an equation is anything with an equals sign in it; a formula is an equation of the form $A={\rm\ stuff}$ where $A$ does not appear among the stuff on the right side.

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    $\begingroup$ I think the second one is a bit different. I think formula is an useful equation or kind of. $\endgroup$
    – Sawarnik
    Jan 28, 2014 at 11:10
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Please down vote me if you wish - but I would say these words are really synonyms to each other. They both express that there is some underlying relation between some mathematical expressions.

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    $\begingroup$ Synonyms? $ax^2+bx+c=0$ is a quadratic equation; $x={-b\pm\sqrt{b^2-4ac}\over2a}$ is the quadratic formula. $\endgroup$
    – Gerry Myerson
    May 10, 2011 at 13:28
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    $\begingroup$ By your definition, Gerry, the quadratic equation is a formula for zero. $\endgroup$
    – Greg L
    Jun 12, 2012 at 4:35
  • $\begingroup$ No downvote. I think that over time the distinction is lost. My math teacher, 35 years ago stated "formulas are used in chemistry, in math we have equations". To this day, the word 'formula' in math seems wrong, but I'd accept it's used commonly. $\endgroup$
    – JTP - Apologise to Monica
    Oct 10, 2013 at 14:48
  • $\begingroup$ @JoeTaxpayer Thanks. Your teacher was right, but that does not exclude formulas in mathematics. I myself use both words. $\endgroup$
    – AD - Stop Putin -
    Oct 10, 2013 at 17:28
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    $\begingroup$ @JoeTaxpayer I my opinion we can use both things as long as we understand each other. $\endgroup$
    – AD - Stop Putin -
    Oct 10, 2013 at 22:00
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You solve an equation, while you evaluate a formula.

By the way, an equation that holds whatever the values of the variables is an identity.

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A formula is an equation that shows the relationship between two or more quantities. It would be the rule or instructions that is used to show the relationship between two or more quantities.

An equation is a problem displayed with numerals or symbols with an equals ($=$) sign included somewhere; usually near the end of the equation. Unless, it is a ratio or division.

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$e=mc^2$ and $f=ma$ are "equations", not normally called "formulas". You wouldn't say the "force formula", but the "force equation". They can have an infinite number of solutions. so i'd say the terms are interchangeable too.

http://en.wikipedia.org/wiki/Force

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A simple answer comes from https://www.bbc.co.uk/bitesize/guides/zwbq6yc/revision/1

For your convenience a succinct explanation from the link is:

A formula:

  1. shows the relationship between two or more variables (e.g. $\frac{9}{5}^{\circ}C + 32 =^{\circ}F$)
  2. is a calculation for a specific purpose (e.g. the conversion from Celsius to Fahrenheit)
  3. is always true, subject to certain conditions, no matter the inputs.

An equation:

  1. will usually have only one variable, though it may appear more than once
  2. will be correct only for certain values (e.g. $2x = 10$ is only true for $x = 5$)
  3. is not always true.

Though I suggest you look at expressions and identities too.

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An equation is a statement that connects two expressions with an $=$ sign and asserts their equality.

An identity is an equation that holds for every variable tuple for which the equation is defined:

  • $(x+y)^2\equiv x^2+y^2+2xy$

A conditional equation holds for some variable tuple(s):

  • $x^2+ky^2=1\quad$ (the parameter $k$ is an arbitrary constant, varying to generate a family of equations)
  • $2x^2+3x-5=0\quad$ (in the context of equation-solving, $x$ is an unknown)
  • In a formula (the rule of a function) like $$V=\pi r^2h,$$ each input tuple returns an output called the subject.

An inconsistent equation holds for no variable tuple:

  • $|2x|=x-1$

Addendum

A constant whose value is unspecified, a parameter and an arbitrary constant all mean the same, and has a unspecified value (unless instantiated) in the context or problem. The first term emphasises the placeholder's fixed value within the context (in contrast with a variable), the second term emphasises its varying value across contexts (in similarity with a variable), while the third term emphasises that its choice of value isn't important.

In the context of a constraint problem, a variable or a parameter or lettered constant can also called an unknown.

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  • $\begingroup$ Imagine that you were writing an elementary book, for example for high school learners, and at the beginning you had a glossary where you wanted to write the definitions for many mathematical words (e.g. numerical expression, algebraic expression, variable, equation, etc.) to help the reader make sense of those words when they appeared throughout the text. In this setting, what would you write to explain/define what a mathematical formula is? Imagine that you wanted to put an entry for formula in a table like this imgur.com/a/1hIc2Rd $\endgroup$
    – fire-bee
    Jun 28, 2022 at 0:10
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    $\begingroup$ Perhaps: A formula is a rule for expressing a subject in terms of some other variable(s); for example, $V=πr^2h$ expresses a cylinder's volume in terms of its radius and height. $\endgroup$
    – ryang
    Jun 28, 2022 at 14:34
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    $\begingroup$ Thanks! I opened a question on math educators, feel free to comment there. matheducators.stackexchange.com/questions/25351/… $\endgroup$
    – fire-bee
    Jun 28, 2022 at 16:00
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An equation is a relationship that defines a restriction. for instance: $ area >= 2*depth*ratio $

In a formula, the equal sign actually means an assignment ($ \leftarrow $): e.g. $ f(x,y) \leftarrow x^2+y^2 $

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In a formula all the variables can be arbitrarily chosen. An equation admits only particular values of non-constant variable/s.

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TL;DR I'd say it really depends on the context.

What I remember in high-school/secondary school:

We were given problems like

Given length and area of a rectangle, find its width.

(Not exactly rectangle. that's of course more primary/grade school. Can be rectangular prism or whatever.)

The 'formula' is $A =wl$.

The 'equation' is what you get when you plug in the given values for $A$ and $l$. So if you have $A=10$ and $l=7$, then the equation is $10=7w$.

At the time, I thought it was very nit-picky/subjective/conventional/contextual. Now, I still do but I have the 10,000+ rep and bachelor's and master's degrees to complain about it.

According to my secondary school teachers, what you provided is a formula, but that's in the context of filling in the blanks of equation and formula in school.

Conclusion: I'd say it really depends on the context. If you define equation as a statement with an equal sign, then every formula with an equal sign is an equation... By the way, it seems on Wikipedia that there are no inequalities that are formulas.

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One way to answer this question has been developed in the first and second courses of U.S. high school algebra.

In the fist course, the following definitions are formally stated in the glossary.

equation: A statement formed by placing an equals sign between two numerical or variable expressions.

For example, $11-7=4$, $5x-1=9$, and $y+2=2+y$ are all equations because they all satisfy the given definition.

The following definition for a formula is also found formally stated in the glossary:

formula: An equation that states a rule about a relationship.

Here are two useful formulas: $A=lw$, the formula for the area of a rectangle; $P=2l+2w$, the formula for the perimeter of a rectangle.

While at the beginning of the the second course, the following sentence summarizes that the relationship is between two or more variables:

A formula is an equation that states a relationship between two or more variables.

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My definitions will focus on mathematics.

In context of mathematics, an equation is a mathematical expression in terms of numbers, variables, operations, and equals sign which is satisfied by some specific values of a desired variable for given values of other variable(s) in it. Example: $$x^3-6x^2+11x-6=0$$ The above example shows a cubic equation in term of $x$ which is satisfied by specific values i.e. $x=1, 2,3$ only.

While a formula is an equation in which a desired variable is expressed explicitly in terms (i.e. explicit function) of other variable(s). Example: The circum-radius $R$ of a triangle having sides $a, b$ and $c$ and area $\Delta $ is given by following expression

$$R=\dfrac{abc}{4\Delta}$$ In the above example, variable $R$ is explicitly expressed in form of other variables or known parameters $a,b$ and $c$ (while $\Delta$ is dependent on $a, b,c$)

In your case, car's fuel efficiency mpg is explicitly expressed in terms of distance traveled and fuel used i.e. ratio of distance and fuel used. Therefore it is best a formula.

NOTE: In context of mathematics, every formula is an equation but an equation is not necessarily a formula.

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    $\begingroup$ This answer is so much misleading. $\endgroup$
    – Deeponjit Bose
    Apr 29 at 11:11

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