SAT Math — circles: radius, sectors, and the equation

Circles carry more avoidable errors than any other shape on the test, and nearly all of them come from four places.

The first is the radius and the diameter. Every formula is built on the radius, and questions routinely give the diameter instead, so the first thing to settle is which one is in front of you. Two questions here work on that directly, and their wrong options are built from using one where the other belonged — the values are close enough that only attention separates them.

The second is squaring. A circumference uses the radius once, an area uses it twice, and a circumference and an area of the same circle look confusingly similar when written in terms of pi. Both are offered in the same option list more than once here, with the explanations pointing out that one is a distance and the other a space. Running the same questions backwards — given an area or a circumference, find the radius — turns the squaring into a square root, and the wrong options are built from halving it instead, which is the same confusion in reverse.

The third is the sector. A sector is simply the fraction of a circle that its central angle is of a full turn, which the questions state outright so that nothing has to be recalled. That single idea covers both arc length and sector area: take the fraction, then apply it to the circumference or to the area as the question requires. A check worth doing is putting the sectors back together, since three one-third sectors must rebuild the whole circle.

The fourth is the equation, and it is where marks are lost most cheaply. Each bracket SUBTRACTS a coordinate of the centre, so a bracket that reads as a plus is subtracting a negative and the coordinate is negative — a plus sign in the equation means a minus sign in the answer. Meanwhile the number on the right is the radius SQUARED, not the radius. Two questions are built on these two facts, one asking for the centre and one for the radius, with wrong options for every way the signs can be mishandled and for taking the right-hand side at face value.

Two more place a point relative to a circle: on it, inside it, or outside it, decided by comparing its distance from the centre against the radius.

Every circle and point here is invented, and nothing is drawn from any official publication.

• Decide whether a given length is a radius or a diameter before choosing a formula
• Tell a circumference from an area when both are written in terms of pi
• Work backwards from an area or a circumference to a radius, taking a root rather than halving
• Treat a sector as a fraction of the whole circle, and apply that fraction to either the circumference or the area
• Read a centre out of a circle's equation, reversing the sign inside each bracket
• Read a radius out of a circle's equation, remembering that the right-hand side is its square
• Place a point on, inside or outside a circle by comparing its distance from the centre with the radius

Written for this catalogue, with no source document. The content is the circle material named in the College Board's own public description of the Geometry and Trigonometry domain — circumference, area, arcs and sectors, and circles in the coordinate plane; every question, option and explanation here is original, and nothing is reproduced from any official publication or released test. Note that the digital test presents much of this material with a figure on screen; here every configuration is described in words and every formula beyond the two basic ones is stated in the question itself. Zestly is an independent study tool. It is not affiliated with the College Board, which owns the SAT, and it is not an exam centre.