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# Circle equations — Basic example

Watch Sal work through a basic Circle equations problem.

## Want to join the conversation?

• Can you put up all the equations, one should memorize, for the SAT?
• yes please. I second this proposal. I can't remember all the formulas.
• Why isn't the answer (5,5)? Because -2 and 8 have exactly 10 lines/spaces separating them, and dividing that by 2 is 5.
• You're right. 5 represents the number of UNITS away the midpoint from each end point.
-2+(5units)=3 and 8-(5units)=3
3 is the midpoint.
• I figured it out before sal even drew the circle ;-; for the first~ I got something right without a mistake
• LOL. Great job Idevanand84.
• how did u know to divide by two?
• That is the midpoint formula. (x1+x2)/2, (y1+y2)/2
• How do you get 2 pi if you went all the way around? I thought if you went all the way around it was 3 pi.
(1 vote)
• By definition, a radian is a "a unit of angle, equal to an angle at the center of a circle whose arc is equal in length to the radius."
Now, the circumference of a circle is 2 pi times radius. Since each radian angle corresponds to an arc of length r, in the angle corresponding to the whole circle equals 2 pi radians.
• how to find the equation?
• You'd need to determine the center of the circle and the radius of the circle. Sal determined in this video that the center is `(3, 5)`. We can then use the Pythagorean Theorem to find the circle's radius. `(2, 0)` is a point of the circle, and its distance from the center `(3,5)` is `sqrt(1² + 5²)` or `sqrt(26)`. The equation of a circle is `(x-u)²+(y-v)²=r²`, where `(u, v)` is the center of the circle and r is the radius of the circle. So we plug them in and find that the equation of this circle is `(x-3)²+(y-5)²=26`!
(1 vote)
• What if you're given the radius and you're supposed to give the equation
(1 vote)
• Well, the equation of a circle is (x-u)²+(y-v)²=r² , where (u, v) is the center of a circle and r is the radius of the circle. So, if you know the center of a circle and the radius of a circle, you can construct the equation of a circle! For example, if you know the radius of a circle is 8, then the equation must be (x-u)²+(y-v)²=64. Here, u and v don't even matter!

I hope this helps!
• at wouldn't be over -2 instead of 2?
• That's a very good question. In this problem, you are using the midpoint formula to find the center of the circle. The midpoint formula is: (x1+x2)/2, (y1+y2)/2. The 2 that you divide by is a constant +2; so, it wouldn't be a -2 because that would change the sign of the quotient.
(1 vote)
• how can i improve my speed in solving problems??
(1 vote)
• You start taking the practice tests and see how fast you go on them@nilathegreat2006
(1 vote)
• What is the general equation for a circle
(1 vote)
• The general equation for a circle is (x-h)^2 +(y-k)^2 = r^2, where (h,k) is the center of the circle and r is the radius
(1 vote)

## Video transcript

- [Instructor] A circle in the xy-plane has a diameter with endpoints at eight comma 10, and negative two comma zero. Which of the following is the center of the circle? So let's think about it. If you have some circle in the xy-plane, so this is my best attempt at drawing a circle, and if you had two endpoints of a diameter, so let's say that this is a diameter right over here. And it looks something like this. So let's say that's our diameter right over there, and if this point was the point negative two comma zero and this was the point eight comma 10, what would be the coordinates of the center of the circle? Well the coordinates of the center of the circle is going to be the midpoint of these two endpoints of a diameter. In fact, the center of a circle is always going to be the midpoint of the two endpoints of the center of a diameter. If the diameter was this right over here, if the diameter was this over here, then once again the center of the circle is the midpoint of that point and that point right over there. So to answer this question we just have to find the midpoint of this point and this point right over there. And to find a midpoint we just have to find the average of the x-coordinates and the average of the y-coordinates. So let's see, this is gonna be eight, the x-coordinate. The average of eight and negative two is eight plus negative two over two. And the average of 10 and zero, we know that's five, but we could say that's 10 plus zero over two. Now let's see. Eight plus negative two is six, divided by two is three. And then ten divided by two is five. Three comma five. And we see it right over there.