The SAT gives a circle as an equation and asks for its center, its radius, or its diameter. When the equation is already in standard form, both facts are readable in seconds, with two sign and square-root details to get right. When it is expanded, you complete the square first.
Standard form is the quantity x minus h squared plus the quantity y minus k squared equals r squared. The center is the point with coordinates h and k, and the radius is r. Two details cause almost all the errors. The coordinates appear with opposite signs from what is written inside the parentheses, and the number on the right is the radius squared, not the radius.
Take the quantity x minus 3 squared plus the quantity y plus 5 squared equals 49. The center is at the point with coordinates 3 and negative 5, because y plus 5 is y minus negative 5. The radius is the square root of 49, which is 7, and the diameter is 14. All three numbers, 49, 7, and 14, will be among the answer choices, so read which one the question wants.
When the equation is expanded, complete the square separately for x and for y. Start with x squared plus y squared minus 6x plus 4y equals 12. Group the terms as x squared minus 6x plus y squared plus 4y equals 12. Half of negative 6 is negative 3, and squaring gives 9. Half of 4 is 2, and squaring gives 4. Add both to each side.
That gives x squared minus 6x plus 9 plus y squared plus 4y plus 4 equals 12 plus 9 plus 4, which is the quantity x minus 3 squared plus the quantity y plus 2 squared equals 25. So the center is at the point with coordinates 3 and negative 2, and the radius is 5. The phrase add to each side is where mistakes happen, since adding only on the left changes the circle.
Some questions never name the center. They ask whether a given point lies on the circle, which you test by substituting the coordinates and checking whether the equation holds, or they give the center and a point on the circle and ask for the equation. In that second case the radius is the distance between the two points, and squaring that distance gives the number on the right without ever taking a square root.