Circles

This skill point has two halves that look unrelated and share one definition. The COORDINATE half asks you to read a circle's centre and radius out of an equation, or to build the equation from a description — and its hard variant hides the circle inside an expanded form that has to be reassembled by completing the square. The CLASSICAL half asks about arcs, sectors, radians, and the angles a chord subtends at the centre and at the rim. Both halves come out of the same sentence: a circle is the set of points at a fixed distance from a fixed point.

College Board publishes the domain weight: Geometry and Trigonometry is about 15% of the Math section, roughly five to seven of the 44 questions, spread across four skill points — area and volume, lines and angles and triangles, right triangles and trigonometry, and this one. What College Board does not publish is how those five to seven questions divide among the four, or how they distribute between the fixed first module and the routed second module. Any specific claim you have read that circles are worth one or two questions per test is a prep-industry estimate reverse-engineered from released material, not an official figure. Plan with it; do not cite it as fact.

One official fact is worth more than any estimate: what the reference sheet actually gives you. It supplies A = πr² and C = 2πr, and it states that a circle contains 360 degrees of arc and 2π radians of arc, and that a triangle's angles sum to 180°. It does NOT supply the equation of a circle, the arc length formula, the sector area formula, the inscribed angle theorem, or the degree-to-radian conversion in usable form. Everything on this page except area and circumference has to be carried in. That asymmetry is why this skill point punishes thin preparation harder than the rest of the domain.

There is a weak path and a strong path through nearly every item here, and the gap is about two minutes. Weak path: an item says a circle has centre (3, −4) and passes through (8, 8), and you start hunting for a formula for a circle through a point. Strong path: the radius IS the distance from the centre to any point on the circle, the horizontal and vertical gaps are 5 and 12, that is a 5-12-13 triangle, so r = 13, r² = 169, and the equation (x − 3)² + (y + 4)² = 169 is written in one line. Same knowledge, different first move.

If you can look at x² + y² − 6x + 10y = 2 and say "centre (3, −5), radius 6" inside twenty seconds, skip ahead to the next block. If any part of that is unfamiliar, this is the highest-value block on the page, and none of it takes long.

The definition, and everything follows from it: a CIRCLE is every point that sits exactly r units from one fixed point. That fixed point is the CENTRE, and r is the RADIUS. The centre is not on the circle — it is the thing the circle is measured from. A DIAMETER is any segment through the centre with both ends on the circle, and it is exactly twice the radius. That factor of two is responsible for more lost points on this skill than any genuine geometry.

The vocabulary the exam uses without defining. A CHORD is a segment joining two points on the circle (a diameter is the longest chord). An ARC is a piece of the circle itself — a curved piece of the rim. A SECTOR is a pie slice: the region between two radii and the arc between them. A TANGENT line touches the circle at exactly one point; a SECANT line cuts through it at two. A CENTRAL ANGLE has its vertex at the centre; an INSCRIBED ANGLE has its vertex on the circle.

π is not a mystery, it is a ratio: divide any circle's circumference by its diameter and you get the same number, roughly 3.14159, every time. That single fact is both formulas. Circumference C = πd = 2πr. Area A = πr². Both are on the reference sheet, so you never need to recall them — but you do need to keep them apart, because one is a length and the other is an area, and the exam builds an answer choice out of each.

Degrees. A full turn is 360°, a half turn 180°, a quarter turn 90°. The MEASURE of an arc, in degrees, is defined to be the measure of the central angle that cuts it off — so a 90° arc is a quarter of the rim regardless of how big the circle is. Arc measure and arc length are different things: measure is an angle and does not care about the radius; length is a distance and scales directly with it. The exam writes items that hand you one and ask for the other.

The coordinate-plane tools this skill leans on. The DISTANCE between (x₁, y₁) and (x₂, y₂) is √((x₂ − x₁)² + (y₂ − y₁)²), which is nothing but the Pythagorean theorem with the horizontal and vertical gaps as the legs. The MIDPOINT of the segment joining them is ((x₁ + x₂)/2, (y₁ + y₂)/2) — average the x's, average the y's. Both appear inside circle items constantly, usually without being named.

COMPLETING THE SQUARE, in one variable, from zero. The expression x² + 10x is not a perfect square, but it is one term short of being one: half of 10 is 5, and 5² is 25, so x² + 10x + 25 = (x + 5)². The rule is always "halve the coefficient of x, then square it", and the number inside the finished parenthesis is that half, not the square. Since the original expression did not contain the 25, you have to keep the equation honest — either add 25 to BOTH sides, or add and subtract it on the same side. Skipping that bookkeeping is the single most common way a circle's radius comes out wrong.

Square roots, and one convention. If r² = 20 then r = √20, which simplifies to 2√5 because 20 = 4 · 5 and √4 = 2. A radius is a length, so the negative root is discarded without comment. Answer choices on this skill are frequently four different radicals, and simplifying yours before you look is what makes them distinguishable.

STANDARD FORM: (x − h)² + (y − k)² = r², a circle centred at (h, k) with radius r. Three readings, and each has a trap sitting on it. The centre's coordinates appear SUBTRACTED, so (x + 4)² means h = −4 — every sign inside a parenthesis is the opposite of the number it names. The right-hand side is r², not r, so a right side of 20 is a radius of 2√5. And nothing here is a function: a circle fails the vertical line test, which is why this equation is never written as y = something and why solving for y is almost always the wrong first move.

GENERAL (expanded) FORM: x² + y² + Dx + Ey + F = 0. Expand any standard form and this is what you get, which is exactly why it is recoverable. Recognise it on sight by two features: an x² and a y² term with the SAME coefficient, and no xy term. If the x² and y² coefficients differ, the shape is not a circle and none of this applies.

The recovery procedure, with the step people drop marked. 1) If the x² and y² terms have a coefficient other than 1, DIVIDE THE ENTIRE EQUATION by it first — this is the step that gets skipped, and skipping it corrupts every number downstream. 2) Group the x terms together and the y terms together, and move the constant to the right-hand side. 3) Complete the square in x: halve the x-coefficient, square it, and add that number to BOTH sides. 4) Do the same in y. 5) Write the two perfect squares on the left and add up the right. What is now on the right is r², and the centre is read off with the signs flipped.

Worth carrying as a shortcut once the procedure is automatic: for x² + y² + Dx + Ey + F = 0 the centre is (−D/2, −E/2) and r² = D²/4 + E²/4 − F. It is the same algebra pre-run, and it is faster when D or E is odd and completing the square would put fractions on the page. Use the shortcut to check the long way, not to replace it — a memorised formula with a sign error in it is worse than four lines of visible work.

The four questions this half of the skill actually asks. What is the centre / radius / diameter? Write the equation given a centre and a point on the circle (the radius is the distance between them), or given the endpoints of a diameter (the centre is their midpoint and the radius is HALF the distance between them). Does a given point lie inside, on, or outside — compare (x − h)² + (y − k)² against r², smaller means inside. And: for what value of some constant is the circle tangent to a line, or does the equation fail to describe a circle at all.

A RADIAN is defined as the angle you get when the arc it cuts off is exactly as long as the radius. It is therefore a ratio of two lengths — arc over radius — which makes it a pure number with no units attached. Going all the way round covers a circumference of 2πr, which is 2π radii, so a full turn is 2π radians and a half turn is π radians. That single equivalence, 180° = π radians, generates every conversion: multiply by π/180 to go from degrees to radians, and by 180/π to come back. The one you want is the one whose unwanted unit cancels.

The angles the exam actually reuses, worth knowing cold in both systems: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 120° = 2π/3, 135° = 3π/4, 150° = 5π/6, 180° = π, 270° = 3π/2, 360° = 2π. If an angle in radians has π in it, its degree measure is almost always one of these, and recognising it saves the conversion entirely.

ONE METHOD covers arc length and sector area, and it never needs a formula you have to recall correctly under pressure. Work out what FRACTION of the whole circle the angle represents — θ/360 if θ is in degrees, θ/(2π) if θ is in radians — and then multiply that fraction by the whole thing you want a piece of. Arc length = fraction × circumference. Sector area = fraction × area. The fraction is identical for both; only the thing being sliced changes.

The radian shortcuts, which are the fraction method with the algebra already done. Arc length s = rθ. Sector area A = ½r²θ. Both are RADIAN-ONLY: putting a degree measure into either one produces a number with no meaning, and it will be an answer choice. There is also a third relation worth having, because it appears in items that give you an arc and want an area without mentioning an angle at all: A = ½rs, sector area equals half the radius times the arc length.

Reading the item to decide which quantity is wanted. "Length of the arc", "distance travelled along the rim", "perimeter of the sector" (which is the arc PLUS two radii — the exam does ask) are all lengths. "Area of the sector", "area of the shaded region" are areas. A length answer carries units like cm; an area answer carries cm². When an item gives you a figure in centimetres and the choices are in cm², the units alone tell you which formula the item wants.

CENTRAL ANGLE = ARC MEASURE. An angle with its vertex at the centre cuts an arc of exactly the same degree measure. That is a definition, not a theorem, and it is why the fraction method works at all.

INSCRIBED ANGLE = HALF THE ARC IT INTERCEPTS. An angle with its vertex ON the circle measures half the arc it opens onto — and therefore half the central angle standing on that same arc. Two consequences do most of the work on the exam. First, every inscribed angle standing on the same arc is equal, no matter where its vertex sits on the rest of the circle. Second, an angle inscribed in a semicircle — one whose two arms end at the two ends of a DIAMETER — intercepts a 180° arc and is therefore exactly 90°. Any item that mentions a diameter and a third point on the circle is handing you a right triangle.

TWO RADII MAKE AN ISOSCELES TRIANGLE. Every radius has the same length, so any triangle with the centre as one vertex and two points of the circle as the others has two equal sides and therefore two equal base angles. This is the quietest and most useful fact on the page: it is how a central angle converts into the angles at the rim, and it turns an apparently under-specified figure into a solvable one.

TANGENT ⊥ RADIUS. A tangent line touches at one point and meets the radius drawn to that point at exactly 90°. Every tangent item on the exam is that right angle plus the Pythagorean theorem. Two more that follow from it: the two tangent segments drawn to a circle from the same external point are equal in length, and the distance from an external point P to the point of tangency is √(d² − r²) where d is the distance from P to the centre.

A CHORD AND ITS PERPENDICULAR. The perpendicular dropped from the centre to a chord bisects it. So the radius, half the chord, and that perpendicular distance form a right triangle: r² = d² + (c/2)², where c is the chord's length and d is the centre's distance from it. This is how the exam asks for the length of the piece a line cuts off a circle without ever using the word chord.

At this point the equation is automatic, the fraction method is automatic, and the inscribed angle theorem is not going anywhere. Six things are still moving points: chords and the perpendicular distance, tangency stated as a condition rather than a fact, general forms carrying an unknown constant, circles meeting other circles, degenerate cases, and a short list of execution errors that survive knowing every rule above.

CHORD LENGTH IS A RIGHT-TRIANGLE PROBLEM. The perpendicular from the centre to a chord bisects it, so the radius (hypotenuse), half the chord, and the perpendicular distance d form a right triangle: c = 2√(r² − d²). The item almost never says "chord" — it says a line, or an axis, cuts the circle, and asks for the length of the segment between the two intersection points. Getting the distance right is where it is won: the distance from a centre (h, k) to the y-AXIS is |h|, and to the x-AXIS is |k|, and reaching for the wrong coordinate produces an answer that is fully in the choice set. If d = 0 the chord is a diameter; if d = r the line is tangent and the chord has length zero.

TANGENCY AS A CONDITION. "The circle is tangent to the x-axis" means the perpendicular distance from the centre to that line is exactly r — nothing more. So a circle centred at (5, −1) is tangent to the x-axis when r = 1 and tangent to the y-axis when r = 5, and an item that leaves a constant in the equation is asking you to set one of those two equalities and solve. The line-and-curve version is the same statement algebraically: substitute the line into the circle and set the resulting quadratic's discriminant to zero, since exactly one intersection point and exactly one solution are the same sentence.

AN UNKNOWN INSIDE THE GENERAL FORM. For x² + y² + Dx + Ey + F = 0 the centre is (−D/2, −E/2) and r² = D²/4 + E²/4 − F, and when one of those letters is the unknown the item becomes an equation in it. This is also where the DEGENERATE cases live, which the exam uses as a correct answer rather than a curiosity: if the right-hand side comes out ZERO the equation describes a single point, not a circle, and if it comes out NEGATIVE no point in the plane satisfies it and the graph is empty. "For what values of k does this equation define a circle" is asking for r² > 0.

CIRCLES MEETING CIRCLES. Compare the distance between the centres, D, with the two radii. D = r₁ + r₂ means externally tangent, touching at one point with each circle outside the other. D = |r₁ − r₂| means internally tangent, one circle touching the other from inside. Between those, they cross at two points; beyond the sum, they miss entirely. And the shortest distance from an external point to a circle is D − r, not D — the point on the circle nearest you sits one radius closer than the centre.

THE UNITS AND FORMAT RULES, which are College Board's own and which cost real points here. Student-produced responses accept integers, decimals, and fractions only — there is no way to type π or a radical — so an item whose answer is 3π must either be multiple choice or be phrased as "the arc length is kπ; what is the value of k". Read that phrasing carefully, because the answer it wants is 3 and the number in your working is 3π. Where a decimal is wanted, the response must fill the entry field: an answer of 2/3 entered as .66 is marked wrong, while .6666 or .6667 is accepted, and a negative sign consumes one of the available characters.

EXECUTION ERRORS, which is where the remaining points actually are. Reporting the diameter when the question asked for the radius, or the radius when it asked for the diameter — both are in the choice set, always. Reporting r² because it was the last number written down. Giving the arc when the item asked for the area of the sector, or the sector's area when it asked for the perimeter of the sector, which is the arc plus TWO radii. Answering in degrees when the item said "in radians", which is a guaranteed distractor whenever the angle is a familiar one. Taking the distance from a centre to the wrong axis. And the quietest one: an item that gives a diameter's two endpoints and gets the centre right, then uses the whole distance between them as the radius instead of half of it, doubling every subsequent number without a single arithmetic error. Before selecting, name the quantity the last line of the question asked for, and check that the number you are about to enter is that kind of thing.