Google’s Free Online Calculator

It’s probably happened to you before: you go to do your math homework and you suddenly realize that you don’t have your calculator handy. What if you have a test tomorrow and you absolutely need your calculator right now to help you study? If you have your smartphone or computer with you, you can easily access Google’s free, built-in calculator, and your problem is solved!

Google has included many special features into their search page that aren’t immediately obvious to the users. For example, you can type in New York weather to get a weather report for New York (instead of getting search results for weather websites). Or, if you search for your favourite sports team, you can get the latest score from their game. Of particular interest to this post, however, is the built-in calculator function.

To access Google’s calculator, all you need to do is type “calculator” or enter the mathematical formula into the search box, and you will be presented a result in an online calculator app. From there, you can make changes to your formula, or clear it and start over again. This calculator even goes beyond basic math functions, by providing buttons for trigonometry, logarithms, exponents, etc. You can even change between Radians and Degrees modes, just like on a real scientific calculator.

This calculator isn’t just limited to being available on desktop browsers. You can access it just the same from mobile browsers on your iPhone or other smartphone. The mobile version has a slight variation to it, in that you are shown a basic calculator when you hold your phone in portrait mode, but flip it horizontally into landscape mode, and it turns into the full-size scientific calculator.

Also, if you have Desktop Voice Search enabled on your Google Chrome browser, all you need to do is click the mic icon, and speak your math equation. Google will interpret your words, and return the calculator to you with the result. Related to this Google calculator is the unit conversion trick you can do in the search box. Simply type in something like “3.25 miles in km” and it will do the conversion for you in the result.

These tricks are great time savers if you find yourself without your calculator, or you just want to get a quick answer to a math question without having to bother going to look for your calculator. And the way website browsers are designed now, it’s takes almost no time to turn on the device and get to a search box, where you can enter your math questions!

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Multiplying Integers – Why Multiplying Two Negatives is a Positive

I have recently been asked to explain why two negative numbers multiply together to product a positive number. This can be a difficult concept to grasp at first, but I would like to try to simplify an explanation. First, I would like to recommend and give credit to David’s website, at Practic-All. In particular, he has posted a document about “Why the Rules for Multiplying Integers Work,” which I highly recommend.

Briefly, to summarize David’s work, you want to think of the two numbers you are multiplying together as being two “groups.” After that, the key to multiplication is to understand that it is a form of repeated addition. That is to say, if you multiply 2 x 3, you are really adding 2 + 2 + 2, or 3 groups of 2.

So, if you have 5 x 2, it is really saying “add 2 groups of 5,” or 5 + 5. (Of course, it could also mean add 5 groups of 2… they both mean the same thing.)
Similary, if you have 10 x 4, it is telling you to “add 4 groups of 10,” or 10 + 10 + 10 + 10. (Or similarly, 10 groups of 4.)

Multiplying two positive numbers is easy. But what about if one of the signs is negative? What does that mean? Well, if you look at it as groups, and remember the key of repeated addition, you will hopefully be able to understand it.

Take 2 x (–5). Using our language, it is telling us to add 2 groups of (–5). The sum is now a larger group equal to (–10).

Another way to look at this is to consider it this way: 2 x (–5) could be the equivalent of saying “take away 5 groups of 2.” Again, this gives us (–10).

Putting all this together, we can hopefully now see why multiplying two negative numbers gives a positive number. (Keep in mind that subtracting a negative number is the same as adding a positive number.) Take the example of (–5) x (–6). We look at it as “take away 5 groups of (–6).” If I write this out, starting with nothing, I could show this as:

0 – (–6) – (–6) – (–6)- (–6)- (–6)

Of course, taking away a negative is like adding a positive, we can rewrite this as:

0 + 6 + 6 + 6 + 6 + 6 = 30

Try to think of multiplication just as a form of adding over and over again, and remember that taking away a negative is the same as adding a positive, and hopefully you will be able to make sense of it. Once again, I highly recommend that you visit David’s site to see multiplying integers visualized in a very helpful way.

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What is Scientific Notation?

So, just what is Scientific Notation?

It is probably one of the first topics you will learn in early physics courses. As such, it is crucial that you understand it and are able to use it. Honestly, without scientific notation, physics problems get INCREDIBLY difficult, and if you don’t get it at first, you should really put in extra effort to figure it out. You won’t regret it… and it really isn’t all that hard. It just sounds crazy.

Scientific Notation is merely a short-hand way of expressing really large or really small numbers. It doesn’t sound all that important, but I will show you how convenient it is.

Let’s think about the measure of time. The length of time it takes the Earth to revolve once around the Sun is a year. We are all familiar with this. We also know that there are 12 months in a year, and 365 days (usually) in a year. Let’s go further though… What about hours? Minutes? Seconds? In a year with 365 days, there are:

365 days * 24 hours/day = 8760 hours
8760 hours * 60 minutes/hour = 525,600 minutes
525,600 minutes * 60 seconds/minute = 31,536,000 seconds

The numbers get pretty big. Why don’t you think about how many seconds there are in 100 years: 3,153,600,000 seconds!

Obviously, you don’t want to have to write down THAT number over and over again, and any numbers you calculate from it, in your equations to solve problems. This is where Scientific Notation comes in.

Scientific Notation basically takes the first non-zero numbers and multiplies that by some factor of 10. Each position in a multi-digit number is represented by 1 power of 10. You have ones, tens, hundreds, thousands, ten thousands, and so on…. and on the opposite side of the decimal, you have tenths, hundredths, thousandths, etc.

So, you can see that:
10 can be written as 1 x 10¹ (one times ten to the power of 1)
100 can be written as 1 x 10² (one times ten to the power of 2)
1000 can be written as 1 x 10³ (one times ten to the power of 3)
and so on…
(If you also recall you rules for exponents, you can see that this pattern continues both up and down… 1 is 10⁰)

Scientific Notation is just expressing things as powers of 10. All you have to do is move the decimal place x number of places so that you have one digit before it, and then multiply your number by 10 to the power of the x places you moved the decimal.

36 = 3.6 x 10¹… because the decimal place moves 1 to the left
189 = 1.89 x 10²… because the decimal place moves 2 to the left
5389 = 5.389 x 10³… because the decimal place moves 3 to the left

What about for bigger numbers?

22000 = 2.2 x 10⁴… move the decimal 4 places to the left. Also note that you don’t need to record zeroes if they are the last digits.

On the other hand:
22001 = 2.2001 x 10⁴… same power, but the zeroes cannot be ignored, because they set the position for the final digit, the 1.

Let’s look back at our example of time:

8760 hours = 8.76 x 10³ hours
525,600 minutes = 5.256 x 10⁵ minutes
31,536,000 seconds = 3.1536 x 10⁷ seconds
3,153,600,000 seconds = 3.1536 x 10⁹ seconds

Now you can appreciate that if you have to rewrite 3.1536 x 10⁹ seconds several times in a calculation, it is simpler to keep track of… and more importantly, there is far less of a chance of accidentally leaving out some of the zeroes and completely getting the wrong answer.

I won’t go into too much detail for extremely small numbers, since it is essentially the same concepts as I have described above. The thing to remember here is that you move the decimal place to the RIGHT this time, and give a negative power of 10. I will, however, leave you with a few examples to work through so that you can hopefully understand it more completely.
0.1 is equal to 1 x 10^(–1) (one times ten to the power of negative 1)
0.01 is equal to 1 x 10^(–2) (one times ten to the power of negative 2)
0.001 is equal to 1 x 10^(–3) (one times ten to the power of negative 3)
and so on…
0.53 = 5.3 x 10^(–1)
0.0687 = 6.87 x 10^(–2)
0.0000873 = 8.73 x 10^(–5)
0.0000000070067 = 7.0067 x 10^(–9)

Another helpful tip is that if your starting number is between 1 and 0 (ie. it is SMALL), it gets a negative power. If it is greater than 1 (ie. it is BIG) it gets a positive power. Positives for big, negatives for small.

Scientific Notation for small quantities is equally as handy as for large numbers. Consider that a virus may be as small as 1 x 10^(–7) m, or that a proton has a mass of 1.7 x 10^(–27) kg. Also imagine what a chemist’s life would be like without Scientific Notation. They would have to write out this proton mass every time they need it in a calculation. It would probably get very tiring, very quickly, and probably with several errors, if they had to write out every time 1700000000000000000000000000 kg. (Of course, with that many digits, one would hope this chemist would include a little bit more precision… but that is a topic for another day!)

As always, let me know if you found this helpful or would like some more clarification!

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The Difference Between Precision and Accuracy

What is the difference between PRECISION and ACCURACY? Upon first glance, many students would say that the two terms mean the same thing. In normal day-to-day usage, you would talk about how precise this is, or how accurate that is, and in both cases you would be comparing “how close” some thing is to the actual thing.

In Math and Science, however, they are slightly different.

ACCURACY has the meaning that I just described: how close a number is to the actual number.

PRECISION, on the other hand, does not refer to how close a number is to an actual number, but rather refers to how many digits and decimal places a number has. (A separate definition for precision also is that it measures how close repeated measurements are to each other, rather than to the actual value.)

As an example, consider a 100 gram block of wood. We are told that this block has a mass of 100 grams. Now, if we put it on balance A, we get a mass of 99.9 grams. When we put it on balance B, we find a mass of 101.350022. We can say that balance A provides us with a more ACCURATE number because it is closer to the known mass of 100 g, but balance B gives us a much more PRECISE number because it displays more decimal places. However, even with the extra decimal places, it isn’t as accurate as balance A. (Maybe there is something wrong with its calibration?)

Do you think that some measurement can be both accurate AND precise? Sure it can.

Let’s suppose we have a bottle of pop. The label says that it is a 1000 mL container. We pour the pop into a graduated cylinder, and determine that there is 998 mL. Since this particular graduated cylinder that we used has no decimal markings, the whole number 998 is as precise as we can get. But that is as precise as the label says. We measure only a 2 mL difference, which isn’t that much (0.2%). So, with this cylinder, we can say that we have a fairly accurate and precise measurement.

If we could determine that the volume in the container was 998.93188 mL, we could say that this number was accurate, and very precise.

Hopefully these examples were able to show you the difference between accuracy and precision. Please comment if you would like some further explanation; this concept always confuses students. I will do my best to clarify.

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What are Real Numbers? Rational, Irrational, Natural, and Integer Numbers

What are Real Numbers? When you are first learning to do math with numbers, you never think about what “kinds” of numbers you use. You just add 2 plus 2, or subtract 10 from 20, and later on begin to multiply (-3) times (-5) (refer to Multiplying Integers here) and other, more complicated functions. What you don’t even realize is that all the numbers that you are playing with actually belong to a certain set of numbers. And this set of numbers is called Real Numbers. Later on, (much later on,) you might begin to study other sets of numbers, such as the Imaginary Numbers… but that is a much different lesson that we’ll save for another day.

So, just what is a Real Number? Quite simply, a Real Number is any number that you can express in a decimal form. They are the numbers that you use everyday in your math homework, and which also get used in laboratories, offices, and industries. Like I said, you don’t even realize that you’ve been using Real Numbers all along.

The set of Real Numbers is often designated with a rather fancy capital R letter, like this: $\mathbb{R}$ .

If we think of our set of Real Numbers as being numbers that can be expressed as a decimal, we can similarly think of them as being represented on a number line. As such, it is easy to see that Real Numbers include all the numbers on the number line, whether they are positive, negative, or zero.

Now, if we continue to think of a number line, we can further break down our set of Real Numbers into further categories.

One category is the Rational numbers, which are any numbers that can be expressed as the ratio of two integers where the denominator is not zero (such as 5 (5/1), 2/3 (0.6666…), or 0.87934 which is really 87934/100000…). The whole numbers, represented by the ticks, are the Integers. And the Integers themselves are are composed of Natural numbers, which are your regular ‘counting numbers’ (0, 1 apple, 2 apples, 3 apples) and their negatives.

Of course, if there are Rational numbers, we must also have Irrational numbers. Irrational numbers cannot be expressed as the ratio of two integers, and so their decimal forms extend forever without repeating. Square roots are sometimes irrational (such as the square root of 2), as well as pi and e (if you haven’t already, you will learn about these special numbers later).

So then, in a nutshell, what is a Real Number? They are a set of numbers that can be broken down into a few categories. Rational numbers are all numbers that can be represented as a simple fraction or repeating decimal. Irrational numbers are all numbers whose decimals continue forever without repeating. Together, the Rational and Irrational numbers fill in our number line completely, and form the set of Real Numbers.

Hopefully this makes sense and answers the question that so many students arrive here looking for: “What are Real Numbers?” Also, I hope it gives you a deeper understanding of the numbers that you have always known how to work with anyways! :)

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Working with Polynomials and Monomials, and How to FOIL

Following up on my post explaining Monomials and Polynomials, here I would like to provide a bit more of an explanation about how to actually work with these terms. In particular, first I am going to show you how to multiply a monomial by a polynomial. Then I will similarly show you how to divide a polynomial by a monomial. And then finally, I will explain the very important concept of FOIL for multiplying to binomials. Nothing too difficult, but all key concepts of algebra. Let’s get to it!

How to multiply a Polynomial and a Monomial. It’s actually very easy, probably way easier than it sounds! Let’s step back a bit first, though, and think about how we multiply regular numbers and expressions. Consider the expression (x + y), and say that we want to multiply that by 3. How do we do that? By now, I think this is very straightforward. We first write it out, like this:

(3)(x + y), or 3(x + y)

Next, we multiply each part of the expression (x + y) by 3, and then we can drop the brackets at the end. This is called the Distributive Property, which refers to how we can remove the brackets if we are multiplying (distributing) the one term with all of the terms within the brackets. I don’t think I need to go into more detail than this for this basic operation, but leave me a comment below if you’d like a bit of help with it. Maybe I’ll do a separate post on the Distributive Property. When we’re done, we have this:

3x + 3y

Now let’s extend this method to some more complicated looking terms. Let’s consider multiplying the monomial 5x with the polynomial (3x + 1). I’ll walk through the steps below, and it hopefully will be easy to follow. All we do is abide by the Distributive Property and simply multiply the single term into the brackets with all the other terms.

5x(3x + 1)
5x(3x) + 5x(1)

Now that we have properly distributed the terms, we just have to simplify it and we’re done. The first term becomes 15x² while the second term becomes simply 5x:

15x² + 5x

And that is all there is to multiplying a monomial with a polynomial. The key is to follow the Distributive Property and multiply things through where appropriate, and then combine like terms if necessary and simplify!

OK, so now hopefully you followed all that. But what about if we want to go the other way? What if we want to start with a polynomial and then divide by the monomial instead? Again, it’s very similar to operations you are already familiar with. In this case, you might compare it to cancelling and reducing terms. Let’s look at an example and you’ll see what I mean.

Let’s say that we want to divide (8x + 4) by 2. Probably pretty simple, right? We just divide the 2 into all of the terms to reduce them, leaving us with the answer (4x + 1). We can think of this as a big fraction, with the (8x + 4) on the top as the numerator, and the 2 on the bottom as the denominator. Because we are adding the 8x and 4, by the laws of fractions, we can separate everything into two separate fractions with the same denominator: 8x/2 + 4/2. Now, it might be even easier to see how to reduce these to get our final answer.

The same logic applies when we want to divide a polynomial by a monomial. Consider the example of dividing (25x³ + 15x² – 5x) by 5x. I’ll write it out as a big fraction first (typing these out is not the most obvious way of demonstrating!).

Now, just like with the example above, we can cancel and reduce things. For cancelling, remember that you can only cancel if you are performing the same operation to each term. Or alternately, if you want to split this into the multiple smaller fractions first before simplifying, if that is clearer to you, then that works as well. Either way, you will find that you arrive at the final answer of:

5x² + 3x – 1

Note that because there is an x term in the denominator, you cancel this out of the numerator for each term by reducing each by x. That is why the powers all reduce by one. The same theory applies to factoring the 5.

Hopefully that explains the two rudimentary concepts of multiplying and dividing monomials and polynomials. Let me know in the comments below if you need any more help!

The last thing I’d like to talk about in this post is FOIL, or how to multiply two binomials. Recall that a binomial is an expression that contains two terms. This is a specific instance of multiplying two polynomials together, though clearly, since polynomials can have many terms, it is best to start simple – so we start here by multiplying two binomials! Larger expressions are not necessarily any harder, though it does become more of a challenge to not mix things up. The key to multiplying any two polynomials is to add up the products of multiplying each term in the first polynomial by each term in the second polynomial.

You have probably heard the expression FOIL when talking about multiplying polynomials, and you will probably see FOIL many times in your studies, so it is best to understand well to what it is referring! FOIL is an acronym that stands for the order of operations that you apply when multiplying two binomials. It only works for binomials, though it still abides by the concept I just outlined for multiplying larger polynomials together. It means FIRST OUTSIDE INSIDE LAST. That probably doesn’t mean much until you see it. It refers to all the pairs of products that you add together when multiplying two binomials.

Here is an example of this:

(2x + 3)(x + 5) = ?

First: 2x * x
Outside: 2x * 5
Inside: 3 * x
Last: 3 * 5

Adding these up gives:
2x² + 10x + 3x + 15

Furthermore, because you are adding polynomials with like terms, you can simplify this expression further:
2x² + 13x + 15

And that’s the final, reduced answer. You can apply the FOIL principle to any two binomials to arrive at their product. When you have more complicated polynomials, such as those composed of 3, 4, 5, or more monomials, you do the same type of thing. What I find easiest is to take the first term of the first polynomial, and multiply it with every term of the second polynomial. Then do the same for the second term in the first one, multiplying with every term in the second one, and so on.

Like this:

(x² + x + 5)(x³ + x² + 1)

First group (multiply x² with all in the second polynomial)

x² * x³
x² * x²
x² * 1

Second group (x)

x * x³
x * x²
x * 1

Third group (5)

5 * x³
5 * x²
5 * 1

Now, you just add up all these terms, and simplify where you can:

(x² * x³) + (x² * x²) + (x² * 1) + (x * x³) + (x * x²) + (x * 1) + (5 * x³) + (5 * x²) + (5 * 1)

(x⁵) + (x⁴) + (x²) + (x⁴) + (x³) + (x) + (5x³) + (5x²) + (5)

(x⁵) + 2(x⁴) + 6(x³) + 6(x²) + x + 5

And that’s it. A little more complicated, but as long as you keep track of what you’re doing and work your way through it, you will arrive at the answer!

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Monomials and Polynomials Explained

When studying algebra, and learning how to perform more complicated rearrangements and calculations, you will frequently see math terms such as “monomials” and “polynomials.” They may sound like some kind of advanced mathematical concepts, but in reality, they are not. In fact, their definitions are really quite simple, and understanding these will help you when working with problems that include them. Understanding the words is important here, because their definitions essentially explain the concepts.

Consider first the term “monomial.” As you probably know already, the mono- part of the word refers to one (or single), just like the word monologue refers to one person speaking, or monotone refers to a sound that has a single pitch. The -nomial part of the word come from the Latin word for name, and refers to terms or expressions (think of them as math names). So, if you put the two parts together, you can see that a monomial is an expression that contains only a single term. Here are a few examples of monomials, and yes they really can be as simple as they look:

3
x
58x
178293xyz
51993x²y³

All of these are single terms, therefore these are all monomials. Any standalone whole number is a monomial, as are variables. You can also combine these to have many parts multiplied or divided, as seen in the examples, but you cannot add or subtract within a term. You also can’t have negative or fractional exponents. Think about a monomial as being anything where all the numbers and letters can be crammed together without having to break it up with an operation. A few things to keep in mind when you work with monomials: a monomial times a monomial is a monomial (eg. x times y is xy), and a constant time a monomial is a monomial (eg. 3 times x is 3x).

So, now that I have defined monomials as being math expressions consisting of only a single term, do you think we have a way of describing expressions that have more than one term? Ha ha! Of course!

Polynomial is the word used to identify a mathematical expression that is composed of multiple terms. Any expression with one term is a monomial, whereas any expression with more than one term is a polynomial. You can break this word down into its components as well, and realize that poly- refers to multiple or many, so now you can easily remember what this word means. Here are some examples of polynomials, and yes these can get very complicated:

x + 3
2x – 10
x2 + 5x – 100
81x⁶ – 43x⁵ + 23x⁴ – 12x³ + 2x² + 6x –1000

All of these expressions are polynomials, and they are all made up of several monomials. Each individual term in these polynomial expressions are monomials on their own. So, as I said to consider monomials as being terms where all the parts are multiplied and crammed together into a single term, as soon as you introduce an operation such as addition, you are creating a polynomial. As you can see on that last example, I ordered all of terms from highest x exponent to lowest x exponent. The highest exponent refers to the degree of the polynomial, which I will discuss in a future post.

While polynomial in a general terms that encompasses all expressions having more than a single term, there are a few other terms that you may see used to describe some specific cases. Binomial refers to polynomials that have exactly two terms, sincebi- refers to two. Similarly, using the prefix tri- gives you the word trinomial for dealing with expressions that have exactly three terms.

So, now that you are familiar with these common math definitions, you can see that they aren’t introducing any additional advanced techniques to your math problems. They are merely words used to describe what you are already used to working with!

In my next post, I will go into a bit more detail on performing calculations with polynomials. For example, how do you divide a polynomial by a monomial, or how do you FOIL two binomials? Follow along, and you’ll learn how to do polynomials problems with ease!

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What are Significant Figures?

Counting Significant Figures is a concept usually introduced to students in beginner science courses, right near the beginning. However, it is a concept that continually confuses people even beyond their school years. This is because, at first glance, significant figures (commonly referred to as “sig figs”) appears to be very similar to rounding, but that is where the difficulty arises. It is more than rounding, and so there are many more rules to follow. It actually describes the amount of uncertainty there is in a value, and is tied to that value’s accuracy and precision. Sig figs are the digits in a value that are considered to be important (significant) as a result of the precision in their measurement. Counting significant figures is a good way to learn about them.

Here are a few basic examples:

A thermometer reads 26.9 degrees Celsius. This is 3 sig figs.
A scale displays a weight of 31.22 g. This is 4 sig figs.
A stopwatch shows 58.778 seconds. This is 5 sig figs.

Did you follow that? The basic rule for counting significant figures that I followed there was that all non-zero numbers are significant. These are simple examples, but there are more rules to consider. The rules that help us identify a sig fig are not difficult, and there are not many, but they are very important to understand. As always, understanding the basics is absolutely essential to being able to move forward. So, let’s learn how to count sig figs!

As I stated above, the first and simplest rule is to consider all non-zero numbers as significant.

123 has 3 sig figs
0.007654 has 4 sig figs

The remaining rules instruct you how to deal with numbers that contain zeroes.

If a number starts with one or more zeroes on the left side, you do not count these as sig figs. You begin counting significant figures from the leftmost non-zero digit.

0.1 has 1 sig fig
0.0045 has 2 sig figs

A zero that is found between other non-zero digits always counts as significant.

101 has 3 sig figs
0.3056 has 4 sig figs

A zero found at the end of a number containing a decimal point does count as significant. However, if the zero is at the end of a non-decimal number, then it may or may not count as significant. Generally, it is safe to count the number of significant figures to the left of the ending zero(es), and then state that there may be up to how many digits you see, depending on the uncertainty of the measurement.

100.0 has 4 sig figs
100. has 3 sig figs
100 has at least 1 sig fig, but could possibly have up to 3 sig figs.
50.30 has 4 sig figs
2.03000 has 6 sig figs
0.0098700 has 5 sig figs
9,885,000 has at least 4 sig figs, but could possibly have up to 7 sig figs.

As I mentioned, the number of significant figures is often a result of the degree of precision in a measurement. However, sometimes the value must be taken in context to find the correct number of sig figs. For example, look at the number 100 above. I said it has at least 1 sig fig but possibly more. If we are talking about a count of something… say, the number of houses on a street, then that value is precise to 3 digits, and it would be fair to say that, in this case, it would have 3 sig figs. Sometimes, this can be noted with a decimal point at the end, without any extra zeros (100.).

And with that, you have now learned about all the rules for counting Significant Figures. They’re not difficult once you practice for a while and recognize where the various rules apply.

Now that we have figured out just what counts as a sig fig, the next step is being able to do math with them – add them, multiply them, etc – and to do that, of course, we have an entirely different set of rules. This is the part that ALWAYS confuses people, so I will do my best to lay it out for you so that it makes sense.

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My Guest Post on the Help Teaching Blog!

I just wanted to post a quick announcement here that my guest post has been published over at the Help Teaching blog! Thanks to Lilia and her staff for giving me the opportunity to publish something on their site, which I hope will be as valuable as the rest of their content that they have to offer. In my post, I put together a list of (what I believe to be) three incredibly useful tools for math educators. I personally use two of the three tools, though that is only because one is PC-based whereas I work on a Mac! I’d love to say more about these tools here, but instead, I will direct you over to the actual post! Please head on over and give it a read, and let me know what you think. Do you use any of these tools as well? Are there any I missed that you think are even better? Let me know in the comments below!

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How to Calculate the Area of a Triangle

Triangle problems are some of the most common types of mathematics problems you will find when first studying geometry and trigonometry. Some might argue that one triangle concept in particular, the Pythagorean Theorem, is one of the most famous and well-known concepts of all! I have gone into detail on this blog about several trigonometry functions that are useful to help solve triangles, but in this post, I am going to talk about the geometric side. In particular, I’m going to show you how to calculate the area of a triangle. There are actually several ways of doing this. This should probably come as no surprise to you, though. There always seems to be more than one way to do things in math!

Probably the easiest way to do this requires that you know only two things about your triangle: the length of its base, and its height. With those numbers, you simply insert them into this equation and solve for the area:
Pretty easy, right? Do you understand why this equation works, though? Take a look at the equation again. You might recognize that it looks familiar, almost like the equation to solve the area of a rectangle – only this time, multiply by a half. So the area of a triangle is equal to half of the area of a rectangle. You can very easily see this simply by drawing any old rectangle, label the sides as “base” and “height”, and then go ahead and draw in a diagonal line that connects two of the corners. You’ve just created two equal triangles, and each triangle is half of the original rectangle! Well, that’s great if you have a triangle that has one side perfectly horizontal, and the other 90 degrees up perfectly vertical. That is literally half of a rectangle. But what about triangles that have more irregular angles?
Like a triangle with 31, 82, and 67 degrees? No right angles there, and it certainly doesn’t look like half of a rectangle. Check this out though. It still works! Orient your triangle so that it’s long side is on the bottom. Then, draw in a line from its highest point straight down to intersect the bottom.
So far, we have just identified its base and its height. Now, for the fun part! Separate your triangle on the vertical line you just drew, and then take your two new smaller triangles and switch sides – put the one on the left over on the right now, but right next to each other, touching at their corner. Now we have a weird structure that has the same base length, and has two vertical walls that have the same height. Now if you just draw in another line to connect the two tops, you can create a rectangle. And, if you were to measure the area of the new triangle created when you connected the tops, you would find that its area is exactly equal to the area of your first triangle, and obviously together they form the complete area of the rectangle.
So, there you have it. Visual proof of the triangle area formula! That’s a fantastic method to use to find the area of the triangle if you know its base and height (or “altitude” as some may call it). But what about if you don’t know its height? The height of right angle triangles is easy because it is equal to one of the sides. But for other triangles, if you know the side lengths, that doesn’t always mean you can immediately determine the height. In that case, there is another formula that you can use, called Heron’s (or Hero’s) formula. You can use Heron’s formula to calculate the area of a triangle where you know the lengths of all of the sides.

Heron’s formula is sometimes referred to as the “irregular triangle calculation method,” and is named after Heron (or Hero) of Alexandria. Heron was an engineer and mathematician in Ancient Greece, who is credited with inventing many thing, including an early steam engine , a windwheel, and a vending machine. (Check out this Wikipedia page for more information on some of his inventions and accomplishments!) His formula for calculating the area of a triangle contains a few steps, but thankfully, they are not all that difficult. The first thing you have to do is to calculate the half-perimeter (or the semi-perimeter, s). You do this by simply summing the three side lengths (a, b, and c) and dividing in half. Once you’ve done that, then you just plug numbers into the following equation:
It might be easier to remember this equation by breaking it down and looking at what each part is. If you consider that the three bracketed terms are really just the difference in side length from the half-perimeter, then all you need to remember is to multiply those three values by the half-perimeter itself, and then square root the whole thing. I know that sounds like a mouthful, but having some degree of understanding about what more complex formulas are based on is always helpful.

Somewhat related to this method is a much simplified version, which is specifically applicable to equilateral triangles. (Of course, we all know that equilateral triangles are special triangles whose sides are equal and all angles are 60 degrees.) It’s a much simpler equation to remember, without needing to make a first calculation like with Heron’s formula. Here is the formula for the area of an equilateral triangle:
The final method that I will describe here uses trigonometry to find the area. You can use this formula on any triangle, provided that you know the length of two of the sides and the angle between them. (Note: there are variations of this formula, depending on what sides and angle are known). It’s fairly easy to remember, as long as you just think about ABC! Here’s the formula:
This goes back to my diagram above, with the inscribed “height” line in the triangle. Using a basic trig identity for sine (SOH!), you can calculate the height using the hypotenuse of one of the smaller triangles and a known angle. Then, you simply take this trig formula and substitute it in for the height term in our original triangle area formula (1/2 base x height). With some simple rearranging, you can come up with our final half absinc formula here!

That’s all the methods that I’m going to discuss here, though you now know four ways of calculating the area of a triangle! Just pick the right methods for your specific problem, and you will only be a few steps away from solving the area!

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