How To Find The Center and Radius of a Circle

in this video we're going to talk about how to find the center and the radius of the circle given the equation shown below so the first thing you need to be familiar with is the standard form for the equation of circle it's minus squared plus minus squared is equal to squared now the center of the circle can be found by this equation all we need to do is identify the values of and and the radius of the circle is simply so what's and what's in this problem is the number in front of without the negative sign so if you see minus 3 here reverse it is equal to positive 3. is the number next to so we have minus four we're going to switch negative four to positive four so is four thus the center of the circle is three comma four so that's how you can find the center of circle if you're given the equation in standard form now to determine the radius we need to focus on this number this is the same as squared so if squared is equal to 25 what is the value of taking the square root of both sides we can get the answer the square root of 25 is 5. so that is the radius of the circle so that's how you can find the center and the radius of the circle given the standard equation now let's work on another example so let's say we have plus 2 squared and then plus minus 4 squared and let's say all of that is equal actually let's make this minus 5 squared and this is all going to be equal to 49 so feel free to pause the video and identify the coordinates of the center of the circle and also the radius so let's calculate and so we see plus 2 next to we're going to reverse it and make it minus 2. there's minus 5 in front of so we're going to change it from -5 to positive 5. so the center of this circle is going to be negative 2 positive 5. so it's very straightforward to find the center of of circle now let's calculate the radius squared is equal to 49 taking the square root of 49 we get that the radius of the circle is 7. so that's all you need to do in order to calculate the coordinates of the center of the circle and the radius as well now you might be wondering these problems can't be that easy and you're right there are difficult versions of this topic like the problem that we're about to go over because sometimes you may not always get the equation of circle in standard form sometimes you'll see it like this and if you do what do you need to do in order to find the center of the circle and the radius as well if you get problem like this what you need to do is you need to put it in standard form so here's what you need to do first bring the variables together i'm going to leave space here and then bring the variables together and then leave space now the negative 11 i'm going to move to the other side it's going to change from negative 11 to positive 11. now what we need to do at this point is something called complete the square to do that we're going to take half of 8 which is 4 and then we're going to square it don't worry about the negative sign during that process half of 6 is 3 so we're going to add 3 squared now whatever you add to the left side of the equation you must also add to the right side of the equation so that the value on both sides remain the same so we've added 4 squared to the left we need to do that on the right side and we added 3 squared on the left so we're going to do the same thing on the right side now we need to factor this expression and here's shortcut method that will help us to do that so whatever variable we see here we're going to write it and then whatever sign we see in front of the middle term we're going to put it here and then it's going to be plus mean then we're going to write 4 and then squared so squared minus 8x plus 16 can be factored as minus 4 times minus 4. this is what is known as perfect square trinomial now let's do the same thing for the next trinomial so whatever variable we see here we're just going to rewrite it and then whatever sign we see for the middle term we're going to write it here and then plus 3 squared so squared plus 6y plus 9 can be factored as plus 3 times plus 3. 4 squared is 16 3 squared is 9. so the last thing we need to do is combine the numbers on the right side 11 plus 9 is 20 and 20 plus 16 is 36 so now we have the equation in standard form so is positive four is negative three so the center is going to be four negative three now squared is 36 and the square root of 36 is 6. so that's going to be the radius of the circle so now you know how to identify the center of the circle and the radius of the circle given the equation of the circle