A formula is a math rule or relationship that incorporates letters to represent amounts that can be changed. These are known as variables and are commonly portrayed as either a, b, c or x. Formulae are used in daily life, from working out areas and volumes to converting units of measurement. Knowing how to use and rearrange formulae are beneficial skills.
Algebraic formula or the way that we represent most of our information in maths is closely related to a previous video I did called Algebraic Expressions. But in general, algebraic formula would refer to the more common equations that we end up up seeing in both math and science and the ways that we'd solve them. Particularly useful in science for when we want to represent information on something like, say, a graph or rearrange a particular formula to get a piece of information. The three topics I want to cover today is one, some of the more common formulas that appear, particularly in science but also in math. Secondly, how to rearrange these formula, particularly the ones that involve squares or one divided by and showing how to do that one line by line.
And lastly, how to determine the gradient of a graph. And critically, how this relates to the actual equation that came from, which is something which I do see some people struggle with, particularly at GCSE level. We're going to jump right into it with some common formula math. So one of the best things about math is that things are always reliable and you'll always see the same equation. So if you can have the time to memorise these things, they'll always serve you well, no matter what you're doing.
The first thing I want to discuss with some common areas and volumes now, a length is something which is just measured as a line. It's just something like this. And that line is going to have a particular distance. In this example, we're just going to call it A. Okay?
If you want to turn this into an area, that means you're fundamentally changing it into a two dimensional shape, which means it's going to turn in something like this. We'll call this lengthier. B. And if you want to find out what this area is, it's going to be the two of them multiplied together, quickly drawing this out just to complete it. If we make it into, say, a rectangle, then that means that we're going to be having this here and the total area is going to be AB.
Okay, I've discussed this for in a previous video, but we're actually going to try and abstract it a bit now because although this is simple for a rectangle for all shapes, what you generally see is that if you want a length, it's always just going to be one measurement of length, which makes sense. So some kind of ranging with x, where x is a value measured in metres area is always going to be something like x squared because you've got two lengths multiplying by one another. And then if we have something like, say, volume, that's just a case of them making this a little bit bigger on the Zaxis. And that's what introduces us to this idea of this being a volume. And the volume of this entire thing would be ABC.
If we keep the pattern, but that means that the volume of a shape is given as some kind of x cubed. And we see that for all kinds of shapes. The two most common shapes that we deal with in maths are a cube, and for a cube we're going to have a length, which I'm going to put in a quotes, which is like how long the side is, and we're going to call it R instead of X, because that's a word that we can use sometimes for the radius and things like that. So R appears a fair bit when doing lengths. Then the area of one face is going to be R squared and then the volume is going to be R cubed.
However, this is why we call it a cube, in fact, because that's where we get the word for it from. And then if we then look to instead a sphere, so change my colour here to a sphere instead. It's got a radius of R. Some of you might be wondering actually why we use radius instead of diameter quite a lot of the time whenever we're discussing circles, especially when we define pi as something which is relevant to the circumference and the diameter. The reason for that is because quite a few of the equations are nicely simplified when it comes to dealing with the radius of objects.
Plus it turns out that because the radius is halfway along something, if an object has some kind of symmetry, we often find that the radius is more useful for describing it than just saying half the diameter over and over again. Now, the surface area of the sphere is a really useful equation that you may not know, and that is equal to four pi R squared. Not necessarily obvious where that would come from. The pi is clearly related to the fact that this is something to do with the circle. But why four appears?
I want you to look up yourselves. And our squared appears because we're dealing with an area which means we're going to be dealing with some kind of square. And then lastly, the volume of a sphere, something which is very common, is four divided by three times pi times R cubed. Spheres are very efficient ways of packing the volume of something such that you minimise the area. So they're very useful shapes in that regard.
That's the reason why we see so many things in nature, having the shape of a sphere, because it allows something to try and minimise its shape as much as possible without having any lines jutting out. So the volume of a sphere appears a lot in physics, and I thought I'd just mentioned that here as a very useful formula to know. Next for common formula is we've got distance, speed, acceleration. We mentioned distance a lot because obviously it's a distance gets used quite a lot in real life. So when we look at distance, we usually describe that as some kind of x sometimes, which is referring to a distance.
Sometimes you might see it written as D for distance or sometimes confusingly S for displacement, which is related to distance. Okay, we then have a few equations. So we say that distance is equal to speed divided by time. That was very wrong. We then see that speed is equal to distance divided by time.
You can cheque the units for that by looking for things like miles per hour or metres per second. The speed is always defined as how much an object moves through an amount of space measured in metres in a particular amount of time T. Then we have acceleration, which describes how much speed changes. So acceleration is equal to whatever the speed is divided by time. And that says how much that is going to change.
Again, very common formula. Lastly, we have this one here of energy. There's a few different formula that we use for energy in science and in math. So the first one we're interested in is kinetic energy, which is equal to half times mass times velocity squared. We've then got gravitational potential energy, which is all about gravity, which is equal to mass times the gravitational constant times height.
And then we have work done which equals force times a distance. These two equations are very similar to one another, but what we end up seeing is that we usually use work done when we're talking about going against air or things like that, or friction. So these are just some common equations. I'd recommend you take a look at your own time and it's certainly going to be useful for you going forward in both math and science. Looking at how we then rearrange some of these.
Very common formula. It depends what you want to rearrange it for. So here's a very common equation which is a modified form of speed equals distance divided by time, where we have acceleration on one side. And let's say we wanted to rearrange this equation in terms of acceleration. So to do that, whenever we rearrange a formula, it's like saying we want to get acceleration on its own.
We want to say A is equal to something, so we got to figure out how to get it on its own. Now, to do that, we're going to have to get A being alone on this side because we need to get rid of this T. But we can't yet divide everything by T because as we know from Bid mass, we end up seeing division happen first. So that means that this U would also be divided by T. So what we're going to do is we're going to subtract U from both sides.
So that means that we're going to have V minus U is equal to U plus at minus you. And that cancels these use. Often this little bit where you cancel is skipped. That's fine. If you feel comfortable, have to do it, that's great.
But I know some students who would rather have every step laid out. And if you're one of those students, that's absolutely fine then. Now that we've got V minus U equals at, when I get T on its own, we have to divide both sides by T. So we get V minus U divided by T is equal to at divided by T. T divided by T is going to give one as a very common result is anything divided by itself gives us one.
And we don't normally write one next to any mathematical formula we use, which means that we get the final result. Acceleration is equal to V minus U divided by T, which is actually another way of saying acceleration is equal to change in speed divided by time. That's just how we might rearrange something that's a bit more simple. However, if we increase the complexity, we have something that looks like this. This is the equation for how long a pendulum takes to swing, and it only actually depends on the length of that particular swing.
The rest of it is just constant. Let's say we wanted to rearrange in terms of L. There's a few steps that we could do to go first. However, again, with Bid Math, we begin with B and then I, which is indices, so we can get square everything. Otherwise it would square everything on both sides as well.
You can begin with that, but in general, it's good to try and get the square root on its own first. So the first thing I'm going to do is divide both sides by two pi. Now, I'm just adding this little bit where I'm writing in what I'm doing here. Now, just to show you what you can also do if you feel like that helps you with your confidence, we have T divided by two pi equals two pi over two pi root L over G. These two pi's cancel as before, and now we're going to square both sides.
When you square both sides, everything needs to be squared. So that means that we're going to have if we just put this sort of, like, square symbol here, we're going to have T squared divided by and then it's four pi times four pi. So it's going to be two pi times two pi. So that means that we get four pi squared. That's going to equal the square root of L or G squared is L or G, which means that lastly, if you want to get things in terms of the length L, we then have to multiply by G.
So let's show that now by multiplying both sides by G, that means that we get T squared g divided by four pi squared equals L. Okay? So you could use this term, the length of a pendulum, if you instead wanted to figure out what G was. In this final step, I'll give you guys a second to think about what you'd have to do instead because it's a little bit more involved. There are two ways that you can take either.
The first one is that we can multiply both sides by G. So we end up getting T squared g over four pi squared equals L as before, and then we multiply by four pi squared. So we get the T squared G equals four pi squared L, and then we divide by T squared. So we get the G is equal to four pi squared L over T squared. That's one way that we could potentially do this using the solar tennis we've done so far.
Alternatively, we can take a bit of a shortcut, but only if you feel comfortable enough to do this. What we could also have done is to get G on its own. We could also have turned into one divided by G. So when we do that, we divide both sides by L. And that means that we get T squared over four pi squared l equals one divided by G.
Now, one divided by G equals T squared divided by four pi squared L. So we can essentially flip both fractions. And that means that we get four pi squared l over T squared equals G over one. And anything divided by one is just itself. These are two ways that you could simplify these more complicated pieces of information.
As the last one, which I'm going to leave for you guys to do is we've got this somewhat unassuming formula down here, which is density is equal to mass divided by volume. But in this case, we're going to combine our skills from both places. What I want you to do is if we say we have the volume of a sphere, which is four divided by three pi R cubed, rearrange this equation for the density of a sphere and trying and get R as the subject. So rearrange and do as best you can there. That's about all the time that we've got for you guys.
So hopefully you've learned a lot more about how to rearrange. Formula as best you can and also some of the skills that will help you in science and math. Take care, and I'll speak to you next time.
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An enthusiastic young teacher who uses his technical wizardry to create fun & engaging lessons on our whiteboard. Lewis can help students of any age or SEN tackle the mysteries of maths.