How do you work out the centre of a circle when you only know 4 points on it's circumference?

You only need 3 points to find the center. Here is a hint. The perpendicular bisector of any chord of a circle goes through the center. If you need more help, I will provide it.

Yes I need more help I got that far anyway

You know the perpendicular bisectors intersect at the center.

Look at the points they gave you. Note that they can be paired in an easy way. How about pairing the ones with the same X, and pairing the ones with the same Y. The perpendicular bisectors for that combination is easy to figure now.
Do a rough graph and you will see why it works. You don’t even need to do the math.
Got it?

draw an arc through the points. complete the circle.

@worriedguy has a good approach. Unfortunately, none of the points have the same y value. You can figure out though from (6,6) and (6,-2) that the x coordinate of the midpoint is the same as x coordinate of the center.

I used Geometer’s Sketchpad to plot the points and draw a circle through 3 of them. The fourth point is also on the circle. This tells me that you did not make a mistake in copying the values of the points and that @worriedguy‘s approach will not work for this problem.

I don’t know what grade you are in. Have you studied equations for lines and were you taught how to find the equation of a line perpendicular to another line? If so, here are the steps.
1. Choose two other points
2. Find the midpoint
3. Find the slope of the line through the two points
4. Compute the negative inverse of the slope. This is the slope of the perpendicular.
5. Find the equation of the line through the midpoint with the new slope.
6. Plug in the x value of the center from the first part into the equation for the line. This gives the y coordinate of the center.

Let me know where I’m going wrong

Slope from -1, -1 to 6, -2

(-2—1)/(6—1) = -1/7
Times that by -1 to get the line parallel to that 1/7

Work out the mid point of -1, -1 to 6, -2
(6+-1)/2=2.5 and (-2+-1)/2 – -1.5

y= 1/7x+b
-1.5=1/7*2.5+b
1.5(1/7*2.5)= -13/7

2=1/7*x-13/7
(2+(13/7))/(1/7)=27 :(

To get the slope of the perpendicular, you need the negative inverse, so the slope is the inverse of 1/7, which is 7.