# How to derive the formula for a circle from scratch

If you’d like to derive the formula for a circle from absolute scratch, then your best option would be to draw a diagram such as the one below:

If you look at this diagram carefully, what you will notice is:

• A circle exists and each point on this circle has the coordinate (x, y).
• The centre of the circle can be found at (a, b).
• The circle has a radius ‘r’.
• The right angled triangles in the diagram each have an adjacent length, opposite length and hypotenuse (r).

Once you’ve prepared a similar diagram, your next aim should be to turn your attention towards the right angled triangles which exist within the circle. You should also think about the many different right angled triangles which could fit within the circle provided they emanate from the centre point (a, b).

The reason I’ve mentioned these right angled triangles is because according to Pythagoras’ theorem, when you have a right angled triangle – its adjacent length squared plus its opposite length squared is equal to the length of its hypotenuse squared:

Now, in this case – the adjacent lengths of the right angled triangles which can fit within the circle on the diagram can be described using the expression:

$\left( x-a \right)$ or $\left| x-a \right|$

The opposite lengths can be described using the expression:

$\left( y-b \right)$ or $\left| y-b \right|$

Also, very interestingly:

• Each of the right angled triangles you can think of has a hypotenuse ‘r’.
• ${ \left( x-a \right) }^{ 2 }={ \left| x-a \right| }^{ 2 }$
• ${ \left( y-b \right) }^{ 2 }={ \left| y-b \right| }^{ 2 }$

When you combine all the information above, what you get is a neat formula which looks like this:

${ \left( x-a \right) }^{ 2 }+{ \left( y-b \right) }^{ 2 }={ r }^{ 2 }$

And it turns out… This is the formula for a circle on the x, y plane, whereby, (a, b) is the centre of the circle and ‘r’ is the length of its radius. How spectacular is that? 🙂

Ok, so now that I’ve offended you – you probably want to leave this page, however, please remember why you got here in the first place. You simply want to add Latex formulas, equations and expressions to your answers so that you can pick up extra “Brainliest” answers and have Brainly.com’s users massaging your ego. Since it most likely is the case, bear with me for one moment and let me explain how you can perform such a feat with a minimum amount of fuss…

Now, I’m assuming you are a Google Chrome user because this post has been written for “Google Chrome Users Only”. If you aren’t a Google Chrome user – then surely you like the sound of my voice, or more specifically my average writing skills – and for that I’ll praise you. It’s not often people spend seconds reading my posts let alone minutes or hours.

Oh, you’re still here? I guess you are a Google Chrome user after all, therefore I’ll hurry up and demonstrate how to add damn Latex code to answers.

You’re probably familiar with the “Chrome Web Store”. If you aren’t, here’s its link: https://chrome.google.com/webstore/category/apps . Now, what you must do when in this store is search for the app “Daum Equation Editor” because this app is spectacular and in a league of its own (the guys who developed this app haven’t paid me to say this). This app will let you produce mathematical formulas, equations and expressions in Latex and copy and paste them into your answer boxes on Brainly.com. Furthermore, you won’t have to produce any Latex code whatsoever to achieve your lifelong ambitions. Life can’t get any better than that can it? …Copying then placing mathematical codes in to answer boxes on Brainly.com.

Assuming you have downloaded this app thanks to this post,  play around with it and explore its degrees of freedom. If the expressions you have produced are satisfying, then copy and paste the relevant Latex code it provides you with (underneath your expressions) in to your beloved answer(s) on Brainly.com. Make sure it sits within the $$…$$ tags which can be produced by clicking on the “paste/edit equation” button that can be found on answer boxes on Brainly’s site.

If you have any questions concerning copying and pasting Latex codes in to your answer boxes, just message me on my Brainly.com profile page or comment on this post.

Bye, bye for now. Have fun answering maths questions. Hehe. 🙂

# How To Come Up With Pythagoras’ Equation

Learn how to come up with Pythagoras’ famous formula – from absolute scratch…

To see the full explanation in writing, please visit the link below:

2 ways to derive Pythagoras’ equation from scratch