SAT Math — circle equations: completing the square to find center and radius

Among the circle skills listed for the digital SAT, the College Board's framework names two that separate a routine question from a hard one: completing the square in an equation representing a circle to find its properties, and using the distance formula in problems about circles. The easy version of a circle question hands over (x − h)² + (y − k)² = r² and asks for the center. The harder version hands over an equation that has already been multiplied out, such as x² + y² + 6x − 4y = 12, and leaves the rest to the student. This material trains that harder version.

The twelve quiz questions work through the moves one at a time and then combine them. Students complete the square to find a radius and then a center, and meet an equation whose x² and y² terms both have a coefficient of 2, where dividing first is the whole trick. They find the constant that gives a stated radius, test which point lies on a circle, and decide where the origin sits relative to a circle whose equation has no constant term. In the other direction, they build the expanded equation of a circle from the endpoints of a diameter, from a center and a point on the circle, and from a translation of a known circle. Two questions ask what an equation says about the graph: how a change in the constant changes the radius, and where a circle tangent to the x-axis must have its center. The last question has students find where a circle crosses the y-axis by setting x equal to zero.

Three questions come with a drawing in the xy-plane: a diameter's endpoints, a circle before it is translated, and a circle with its center and one of its points. The drawings show the setup only; the equation still has to be written. The distractors are the classic slips: r² reported as r, the center's signs left unreversed, a coefficient not halved, the completing terms added to one side only, a diameter used as a radius. Each explanation says which slip produces which wrong answer.

The flashcards cover standard and expanded form, the two steps of completing the square, reading the center and radius, the midpoint and distance formulas, tangency to an axis, and the sign test for a point inside or outside a circle.

The material offers a quiz and flashcards. It is practice written by Zestly and is not affiliated with or endorsed by the College Board.

• Complete the square in x and y to find the center and radius of a circle given in expanded form
• Divide out a common leading coefficient before completing the square
• Write a circle's equation from a diameter's endpoints or from a center and a point, in standard and expanded form
• Describe how translations and changes to the constant affect a circle's equation and graph
• Decide whether a point lies inside, on or outside a circle and find where a circle meets an axis

Practice material written by Zestly, based on the College Board Assessment Framework for the Digital SAT Suite (version 3.01, August 2024), Geometry and Trigonometry domain, skill Circles (equations of circles, completing the square, distance formula).