A Math type (3.0% of the bank) with two distinct halves. Covers the standard equation of a circle and reading its centre and radius with the signs flipped, completing the square to get there from an expanded equation, arcs and sectors as fractions of a whole turn, converting between degrees and radians, and the difference between an arc length and a sector perimeter — with six worked examples, four traps and three checks.
Subject: SAT Prep · 61 slides · applied lesson
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Title
SAT Math · Type 16 of 19
3.0% of the question bank — 50 of 1675 questions
Objectives
Two topics share this label. One is the equation of a circle, where the work is completing the square and the reading is a centre and a radius. The other is arcs and sectors, where the work is taking a fraction of a whole circle. They look like one topic and behave like two, so it is worth knowing which half you are in before you start.
One instruction covers both halves: get to the standard form, then read the answer off it. For an equation that means completing the square; for an angle it means a fraction of 360 or of 2 pi.
Naruhodo Tutoring — SAT Math question bank export (sat-question-index.json) — 50 tagged questions of this type in the site's bank
Section
Section 1
Concept
Two topics share this label. One is the equation of a circle, where the work is completing the square and the reading is a centre and a radius. The other is arcs and sectors, where the work is taking a fraction of a whole circle. They look like one topic and behave like two, so it is worth knowing which half you are in before you start.
You will see it phrased in these ways:
Recognising which half you are in takes a second: an equation means algebra and completing the square; an angle means a fraction of the whole circle.
Picture it
The same three-panel card as the survey deck, so the shorthand carries over: what identifies it, what you write first, and what is built to catch you.
Figure (svg): Circles: the tell, the move, and the trap
Both items in the red panel are reading errors rather than method errors. The algebra is usually right and the final extraction is what fails.
Prediction
Recognition first, because it decides which toolkit applies.
Predict first
Which of these is the equation of a circle?
Correct: x squared plus y squared equals 36
Why: A circle needs both variables squared, both with the SAME coefficient, and added rather than subtracted. The second subtracts, which makes it a hyperbola. The third has different coefficients, 1 and 2, which stretches it into an ellipse. The fourth has x to the first power, so it is a parabola opening sideways. Equal coefficients on two added squares is the complete test.
Concept
Two halves, and they need entirely different methods.
| variant | what you do | which half |
|---|---|---|
| Read a centre and radius | read off standard form, flipping signs | the equation half |
| Complete the square | convert an expanded equation to standard form | the equation half |
| Arc length or sector area | take the fraction of the whole circle | the geometry half |
| Degrees and radians | convert using 180 degrees equals pi radians | the geometry half |
The two halves rarely appear in the same question, so identifying which one you are in tells you immediately whether to reach for algebra or for a proportion.
Definition probe
Standard form is three facts written compactly.
Sort into buckets
In (x minus 3) squared plus (y plus 2) squared equals 49, what does each part give?
Discrimination
Three different quantities from the same slice.
Sort into buckets
Which quantity does each describe?
Warm-up
Try it, and count how many things you had to flip or root.
Discussion prompt
What are the centre and radius of (x plus 4) squared plus (y minus 1) squared equals 36?
Hint: The standard form uses minus signs inside both brackets.
Answer:
Centre (negative 4, 1) and radius 6.
The x-coordinate: x plus 4 is really x minus negative 4, so h is negative 4. The sign flips.
The y-coordinate: y minus 1 gives k equals positive 1 directly.
The radius: the right-hand side is r squared, so r is the square root of 36, which is 6 — not 36.
Three separate readings, and each is a place students lose the question: a sign, another sign, and a square root.
Pattern
Three steps, and the first decides which half of the type you are in.
Decide which half: an equation means algebra, an angle means a fraction.
Why: The two halves share nothing but the shape. Knowing which you are in prevents reaching for the wrong tool entirely.
For an equation, get to standard form — completing the square if it is expanded.
Why: Everything a circle equation can be asked about is visible in standard form and hidden in any other.
Read the answer off, flipping the signs of h and k and taking the square root of the right-hand side.
Why: Those three operations are the whole extraction, and each is a designed trap.
For the geometry half, step 2 becomes: write the angle as a fraction of 360 degrees or 2 pi radians, then multiply by the whole-circle quantity.
Section
Section 2
Concept
A circle with centre (h, k) and radius r has equation (x minus h) squared plus (y minus k) squared equals r squared.
| equation | centre | radius |
|---|---|---|
| (x minus 3) squared plus (y minus 5) squared equals 16 | (3, 5) | 4 |
| (x plus 2) squared plus (y minus 7) squared equals 9 | (negative 2, 7) | 3 |
| x squared plus y squared equals 25 | (0, 0) | 5 |
| (x plus 1) squared plus (y plus 6) squared equals 50 | (negative 1, negative 6) | root 50 |
Every circle-equation question is answered from this one line. Reading it correctly is three operations, and all three are tested.
Concept
When an equation is expanded, group the x terms and the y terms and complete each square separately.
The constants you add must go on both sides. Adding them only on the left changes the equation and moves the circle.
Prediction
The standard form subtracts.
Predict first
What is the centre of (x plus 5) squared plus (y minus 3) squared equals 16?
Correct: (negative 5, 3)
Why: The form is (x minus h) squared, so x plus 5 is x minus negative 5, giving h equals negative 5. And y minus 3 gives k equals positive 3 directly. Only the bracket with the plus sign flips. Reading the numbers as written gives the second choice, which is the most common error and is always offered.
Concept
Take the angle over 360 degrees, or over 2 pi radians, then multiply by the whole-circle quantity.
| quantity | whole circle | for an angle of theta degrees |
|---|---|---|
| circumference | 2 pi r | arc length is theta over 360 times 2 pi r |
| area | pi r squared | sector area is theta over 360 times pi r squared |
| a 90 degree slice | one quarter | a quarter of each |
| a 120 degree slice | one third | a third of each |
Compute the fraction once and use it twice. A question asking for both the arc and the sector area needs only one division.
Prediction
It is r squared, not r.
Predict first
What is the radius of (x minus 1) squared plus (y plus 4) squared equals 81?
Correct: 9
Why: The right-hand side is r squared, so the radius is the square root of 81, which is 9. Answering 81 skips the square root entirely — the single most common circle-equation error. The 40.5 halves it as though it were a diameter, which confuses two different operations.
Concept
The boundary of a sector is the arc plus two straight edges, each equal to the radius.
Trace the boundary of the shape with your finger. You cross three edges on a sector, and only one of them is curved.
Concept
180 degrees equals pi radians, and every conversion follows from that one equivalence.
| degrees | radians | fraction of a circle |
|---|---|---|
| 360 | 2 pi | the whole circle |
| 180 | pi | one half |
| 90 | pi over 2 | one quarter |
| 60 | pi over 3 | one sixth |
| 45 | pi over 4 | one eighth |
| 30 | pi over 6 | one twelfth |
If a radian answer does not simplify to a recognisable fraction of a circle, check the conversion before trusting it.
Prediction
Angle over 360.
Predict first
A circle has circumference 24 pi. What is the length of a 45-degree arc?
Correct: 3 pi
Why: 45 out of 360 is one eighth, and one eighth of 24 pi is 3 pi. The second choice divides by the angle rather than taking the angle out of 360, which is the standard setup error. The 6 pi would be a 90-degree arc, a quarter rather than an eighth.
Concept
Both the area and circumference formulas take the radius, so halve any diameter first.
Write the radius on its own line the moment a diameter appears, exactly as in type 8.
Concept
Substitute the point's coordinates; if the two sides balance, the point is on the circle.
This is the circle version of testing a point in an inequality, and it is just as quick.
Prediction
Through 180 equals pi.
Predict first
What is 135 degrees in radians?
Correct: 3 pi over 4
Why: Multiply by pi over 180: 135 times pi over 180 is 135 pi over 180, which simplifies to 3 pi over 4. The sanity check works too: 135 degrees is three eighths of a circle, and three eighths of 2 pi is 3 pi over 4. The third choice inverts the fraction, giving an angle of about 240 degrees instead.
Two truths and a lie
Three of these circle statements are correct. The one left standing is false.
Eliminate the wrong options
Which statement is FALSE?
Survives elimination: c
Why: The right-hand side is r SQUARED, not r. An equation ending in 25 describes a circle of radius 5, not 25. This single step is skipped more often than any other on the type, and the un-rooted value is offered as a choice every time — which makes it easy to select an answer that required no work at all.
Check
Three readings, three chances to slip.
Check your understanding
What are the centre and radius of (x plus 7) squared plus (y minus 1) squared equals 64?
Answer: A
Why: The bracket x plus 7 is x minus negative 7, so h equals negative 7. The bracket y minus 1 gives k equals positive 1. The right-hand side 64 is r squared, so the radius is 8.
Each distractor isolates exactly one of the three readings. That is how this type is built, and it is why doing all three deliberately matters.
Section
Section 3
Worked example
Find the centre and radius of (x minus 6) squared plus (y plus 2) squared equals 100.
Figure (svg): A circle centred at six, negative two with radius ten
Read h: the bracket is x minus 6, so h equals 6 with no flip needed.
Why: The standard form already subtracts, so a minus sign gives the coordinate directly.
Read k: the bracket is y plus 2, which is y minus negative 2, so k equals negative 2.
Why: A plus sign inside the bracket means a negative coordinate.
Read r: the right-hand side is 100, which is r squared, so r equals 10.
Why: The square root is required; 100 is not the radius.
Testing a point one radius from the centre confirms all three readings at once, and takes about five seconds.
Verify: check a point: (16, negative 2) should be on the circle, since it is 10 to the right of the centre.
Why: Substituting gives 100 plus 0, which equals 100, so the point does lie on it.
Answer: centre (6, negative 2), radius 10
Worked example
Find the centre and radius of x squared plus y squared minus 6x plus 8y equals 11.
Figure (svg): A circle centred at three, negative four with radius six
Group: (x squared minus 6x) plus (y squared plus 8y) equals 11.
Why: The two variables complete separately, so grouping them first keeps the work organised.
Complete the x square: half of negative 6 is negative 3, and squaring gives 9. Complete the y square: half of 8 is 4, and squaring gives 16.
Why: Halve the coefficient and square it — the standard completion step.
Add both to BOTH sides: (x minus 3) squared plus (y plus 4) squared equals 11 plus 9 plus 16, which is 36.
Why: Whatever is added to the left must be added to the right, or the equation describes a different circle.
The most common failure here is adding 9 and 16 to the left only. The right-hand side must grow by exactly the same amount.
Verify: check by expanding back: (x minus 3) squared gives x squared minus 6x plus 9, and (y plus 4) squared gives y squared plus 8y plus 16.
Why: Subtracting the 25 from both sides recovers the original equation, confirming the completion.
Answer: centre (3, negative 4), radius 6
Worked example
A circle has radius 9. Find the arc length and the sector area for a 40-degree slice.
Figure (svg): A forty-degree sector of a circle of radius nine
Find the fraction: 40 over 360 is one ninth.
Why: The same fraction applies to both the arc and the area.
Arc length: one ninth of the circumference. The circumference is 2 pi times 9, which is 18 pi, so the arc is 2 pi.
Why: The arc is that fraction of the circumference.
Sector area: one ninth of the area. The area is pi times 81, which is 81 pi, so the sector is 9 pi.
Why: The same fraction applied to the whole area.
Computing the fraction once and reusing it saves half the work. Nine slices of 40 degrees make a full circle, which is a useful check.
Verify: check the units differ: 2 pi is a length and 9 pi is an area.
Why: The arc is measured in linear units and the sector in square units, as they must be.
Answer: arc length 2 pi; sector area 9 pi
Step zero
Before reaching for any method.
Discussion prompt
A circle question appears. What single question tells you which method to use, and what are the two answers?
Hint: The two halves of this type share nothing but the shape.
Answer:
Ask: does the question involve an EQUATION or an ANGLE?
An equation — x squared and y squared appearing — means the algebra half. Get to standard form, completing the square if necessary, then read off the centre and radius.
An angle — degrees, radians, a slice, an arc — means the geometry half. Write the angle as a fraction of the whole turn and multiply by the whole-circle quantity.
They almost never appear together, so the decision is quick and it saves you reaching for completing the square on a sector question.
The one thing both halves share is that the radius must be extracted carefully: rooted from r squared in one half, and halved from a diameter in the other.
Worked example
For the same circle of radius 9 and the same 40-degree slice, find the perimeter of the sector.
Figure (svg): The same sector, with its two straight edges emphasised
The boundary consists of the arc and two radii.
Why: Tracing the edge of the shape crosses the curved arc and both straight sides.
The arc is 2 pi, from the previous example.
Why: The curved portion was already computed as one ninth of the circumference.
Add the two radii: 2 pi plus 9 plus 9, which is 2 pi plus 18.
Why: Each straight edge is exactly the radius.
The difference between this answer and the previous one is exactly 18. If a question offers both, the deciding word is arc versus perimeter.
Verify: check numerically: 2 pi is about 6.28, so the perimeter is about 24.28, comfortably more than the arc alone.
Why: The perimeter must exceed the arc by exactly 2r, which is 18 here.
Answer: 2 pi + 18
Worked example
Convert 5 pi over 6 radians into degrees, and say what fraction of a circle it is.
Figure (svg): A conversion table from radians to degrees with the target row highlighted
Multiply by 180 over pi so the pi cancels: 5 pi over 6 times 180 over pi.
Why: Choosing the fraction that cancels the unwanted unit is the same units method as type 9.
The pi cancels, leaving 5 times 180 over 6, which is 900 over 6, or 150 degrees.
Why: Straightforward arithmetic once the units are arranged.
As a fraction: 150 over 360 is five twelfths.
Why: Dividing by the full turn gives the fraction of the circle.
Working the fraction in radians directly — angle over 2 pi — avoids the conversion entirely, and is usually faster when the angle already has pi in it.
Verify: check against 2 pi: 5 pi over 6 divided by 2 pi is 5 over 12.
Why: The radian route gives the same fraction as the degree route, confirming the conversion.
Answer: 150 degrees, which is five twelfths of a circle
Faded example
From memory. Three readings from one line.
Fill in the blanks
The standard form is (x minus h) squared plus (y minus k) squared equals r squared. In (x plus 7) squared, the centre's x-coordinate is negative 7. If the right-hand side is 49, the radius is 7. And to convert an expanded equation to standard form you complete the square.
Why: Blanks (a) and (c) are the same trap seen twice: the right-hand side is r squared and needs a root. It is the single most skipped step on this type, precisely because the un-rooted number is sitting right there in the equation.
Worked example
For the circle (x minus 2) squared plus (y plus 1) squared equals 25, where does the point (5, 2) lie?
Figure (svg): A number line comparing the substituted value with r squared
Substitute the point: (5 minus 2) squared plus (2 plus 1) squared.
Why: Testing a point means putting its coordinates into the left-hand side.
Compute: 3 squared plus 3 squared, which is 9 plus 9, or 18.
Why: Both brackets are evaluated before squaring.
Compare with r squared, which is 25. Since 18 is less than 25, the point is inside the circle.
Why: A smaller value means the point is closer to the centre than the radius.
One substitution answers on, inside or outside. There is no need to compute the distance itself unless the question asks for it.
Verify: check the distance: root 18 is about 4.24, which is less than the radius 5.
Why: The point is 4.24 from the centre and the radius is 5, so it lies inside.
Answer: inside the circle
Fill the middle
One fraction, two quantities.
Fill in the blanks
A sector of theta degrees is theta over 360 of the whole circle. Its arc length is that fraction times 2 pi r. Its area is that fraction times pi r squared. And its perimeter is the arc plus 2r.
Why: The same fraction serves both the arc and the area, which is why computing it once is efficient. The last blank is the one that separates two answers the SAT often offers together — an arc length and a sector perimeter differ by exactly 2r.
Estimation
A fraction of a known whole bounds the answer.
Predict first
A circle has area 100 pi. Roughly what is the area of a 30-degree sector?
Correct: about 8 pi
Why: 30 out of 360 is one twelfth, and one twelfth of 100 pi is about 8.3 pi. The answer 30 pi mistakes the angle for the fraction, and 70 pi would be the part remaining rather than the slice. Having the fraction in mind first — a twelfth is a small slice — makes the two large answers visibly implausible before any arithmetic.
Check
Add to both sides.
Check your understanding
The equation x squared plus y squared plus 4x minus 10y equals 7 describes a circle. What is its radius?
Answer: A
Why: Complete the squares: half of 4 is 2, squared gives 4; half of negative 10 is negative 5, squared gives 25. Adding both to each side gives (x plus 2) squared plus (y minus 5) squared equals 7 plus 4 plus 25, which is 36. So the radius is the square root of 36, which is 6.
Three of the four choices correspond to a specific incomplete step. Expanding your standard form back out catches all three.
Section
Section 4
Trap
The trap. The equation ends in equals 36, and you report a radius of 36.
The right-hand side is r squared, so the radius is 6.
The wrong answer requires no work at all, which is exactly why it is tempting under time pressure — the number is already written in the equation.
The fix. Say the standard form aloud as you read: equals r SQUARED.
Then take the square root as a deliberate separate step before writing the radius down.
Trap
The trap. For (x plus 5) squared plus (y minus 2) squared, you report the centre as (5, 2).
The form subtracts h, so x plus 5 means h equals negative 5. The centre is (negative 5, 2).
Only the bracket containing a plus sign flips, which makes it easy to flip both or neither.
The fix. Rewrite any plus inside a bracket as minus a negative, at least mentally.
x plus 5 becomes x minus (negative 5), and the coordinate is then visible directly.
Error analysis
A student converting to standard form. The left side is right and the right side is not.
Annotate
On: \( x^2 - 6x + y^2 + 8y = 11 \;\Rightarrow\; (x-3)^2 + (y+4)^2 = 11 \)
Completing the square is the one place on this type where the algebra genuinely can go wrong. Expanding back is a ten-second check and it is decisive.
Trap
The trap. You add 9 and 16 to complete the two squares, and leave the right-hand side unchanged.
The equation now describes a different circle — one with a radius smaller by exactly what you failed to add.
Every subsequent reading is correct, and about the wrong circle.
The fix. Adding to one side of an equation requires adding to the other.
Write the two constants explicitly on the right as you add them on the left.
Elimination
A circle has centre (negative 3, 4) and radius 5.
Eliminate the wrong options
Which equation describes it? Three can be eliminated by reading the signs and the right-hand side.
Survives elimination: a
Why: A centre at (negative 3, 4) requires x minus negative 3, which is written x plus 3, and y minus 4 written directly. A radius of 5 gives a right-hand side of 25. Each wrong answer isolates one of the three readings — the x sign, the y sign, and the squaring — which is exactly how the distractors on this type are built.
Trap
The trap. Asked for the perimeter of a sector, you compute the arc and stop.
The boundary of a sector includes the two radii, so the perimeter is the arc plus 2r.
The arc-only answer is always offered, and it is the answer to the adjacent question.
The fix. Trace the boundary of the shape and count the edges you cross.
A sector has three: one arc and two radii.
Counterexample
A claim about what makes an equation a circle.
Discussion prompt
A student says: any equation with x squared and y squared in it is a circle. Give two counterexamples.
Hint: Consider the coefficients and the sign between the terms.
Answer:
Counterexample 1: x squared minus y squared equals 9. The terms are SUBTRACTED, which makes this a hyperbola rather than a circle.
Counterexample 2: x squared plus 4y squared equals 16. The coefficients differ, 1 and 4, which stretches the shape into an ellipse.
The full condition for a circle: both variables squared, both ADDED, and both with the SAME coefficient.
A third edge case worth knowing: x squared plus y squared equals 0 has only one solution, the point (0, 0) — a circle of radius zero, sometimes called a degenerate circle.
And x squared plus y squared equals negative 4 has no solutions at all, since a sum of squares cannot be negative. Completing the square sometimes produces exactly this, and it means no circle exists.
Edge cases
The method converts any expanded circle equation. What can go wrong with the result?
Discussion prompt
After completing the square, the right-hand side comes out as zero, or as a negative number. What does each mean?
Hint: The right-hand side is r squared.
Answer:
If it comes out ZERO, the radius is zero. The equation is satisfied only by the centre itself, so the graph is a single point rather than a circle.
If it comes out NEGATIVE, there is no graph at all. A sum of two squares cannot be negative, so no real point satisfies the equation.
Neither result means you made a mistake — they are legitimate outcomes, and the SAT occasionally asks about them.
A further complication: if the x squared and y squared terms have a coefficient other than 1, divide the whole equation through by it BEFORE completing the square. Completing on 2x squared plus 2y squared without dividing first gives the wrong constants.
The check that covers all of these: expand your standard form back out and confirm it reproduces the original equation exactly.
Check
Read which quantity is wanted.
Check your understanding
A circle has radius 12. What is the PERIMETER of a 60-degree sector?
Answer: A
Why: 60 out of 360 is one sixth. The circumference is 2 pi times 12, which is 24 pi, so the arc is one sixth of that, which is 4 pi. The perimeter adds the two radii, 12 plus 12, giving 4 pi plus 24.
Choice B is the designed trap and the difference is exactly 2r. Underline whether the stem says arc or perimeter before computing.
Section
Section 5
Matching
Six things you might be asked, six places to find them.
Match the pairs
Why: The first three rows are the algebra half and the last three are the geometry half. Notice that only one of the six involves any real algebraic work — the rest are readings and substitutions, which is why the type is faster than it looks.
Sorting
Which describe circles?
Sort into buckets
Is it a circle?
Item (e) is the one worth noticing: equal coefficients other than 1 still give a circle, but you must divide through before completing the square or reading the radius.
Comparison
Fill the blanks from memory.
Comparison matrix
| the equation half | the arcs and sectors half | |
|---|---|---|
| What you are given | an equation with x squared and y squared | a radius and an angle |
| The main technique | completing the square | taking a fraction of the whole circle |
| The classic error | reading r squared as r | giving an arc when a perimeter was wanted |
| The check | expand back and compare | count the edges of the boundary |
The two columns share no technique at all, which is why identifying the half first is worth the second it takes.
Trade off
Fill in when each is easier.
Comparison matrix
| working in | easier when | the conversion |
|---|---|---|
| Degrees | the angle is given in degrees | fraction is angle over 360 |
| Radians | the angle already contains pi | fraction is angle over 2 pi |
| Converting first | the answer must be in the other unit | multiply by pi over 180, or 180 over pi |
| Not converting at all | you only need the fraction of a circle | none — the fraction is unit-free |
The bottom row is the underused option. If the question only needs a fraction of the circle, the units cancel out of the calculation entirely and no conversion is required.
Real world
One minute on why radians exist at all.
Discussion prompt
Degrees are familiar and radians look awkward. Why does mathematics prefer radians?
Answer:
Because a radian is defined by the circle itself. One radian is the angle that cuts an arc equal in length to the radius, so the definition needs no arbitrary number.
360 is arbitrary — it comes from Babylonian astronomy and the convenience of its many divisors. Nothing about a circle requires it.
In radians the formulas get simpler: arc length is just r times theta, and sector area is one half r squared theta. No fractions of 360 anywhere.
And calculus requires them. The derivative of sine is cosine only when the angle is measured in radians; in degrees an awkward constant appears.
On the SAT this matters practically: if an angle is already given with a pi in it, working in radians avoids a conversion, and the fraction of the circle is simply the angle over 2 pi.
Ranking
Each equation describes a circle. Order by radius.
Put in order
Why: Taking square roots of the right-hand sides: (b) has radius 2, (a) has radius 3, and (c) has radius 7. For (d), divide through by 2 first to get x squared plus y squared equals 25, giving radius 5. So the order is 2, 3, 5, 7. Item (d) is the instructive one: failing to divide through would suggest a radius of root 50, about 7.07, and would put it in the wrong place.
Concept
This type is 3.0 per cent of the section and splits into two halves that need separate practice.
| session | what you do | why |
|---|---|---|
| 1 | Read the centre and radius from fifteen equations already in standard form. | Makes the three readings — two signs and a root — automatic. |
| 2 | Complete the square on ten expanded equations, expanding each answer back to check. | The only genuinely algebraic part of the type, and the check is decisive. |
| 3 | Fifteen arc and sector questions, computing the fraction once and using it twice. | Builds the proportion habit and separates arc from perimeter. |
| 4 | Ten conversions between degrees and radians, plus five point-position questions. | Two small variants that appear regularly and are quick once practised. |
Naruhodo Tutoring — SAT Math question bank export (sat-question-index.json) — filter the bank to this skill tag — 50 questions, roughly a third of each difficulty
Explain it to yourself
Close the deck.
Discussion prompt
Without looking, write the standard form of a circle and state the three things you read from it, including what has to be flipped or rooted.
Hint: Three readings, three operations.
Answer:
The form: (x minus h) squared plus (y minus k) squared equals r squared.
The centre is (h, k), with the signs flipped from what appears inside the brackets — x plus 5 means h equals negative 5.
The radius is the square root of the right-hand side, since that side is r squared rather than r.
To reach this form from an expanded equation, complete the square in x and in y, adding both constants to BOTH sides.
If you stated all three operations — two sign flips and one square root — you have the reading the distractors are built around.
Explain it
Two minutes, out loud.
Discussion prompt
A friend keeps giving the radius as the number on the right of the equation. How do you fix it so it stays fixed?
Answer:
Start with a circle they can picture. x squared plus y squared equals 25, centred at the origin. Ask them to name a point on it.
They will probably say (5, 0). Check it: 25 plus 0 equals 25. Correct — and that point is 5 from the centre, so the radius is 5, not 25.
Name the reason: the equation is really about the distance formula. Distance squared equals r squared, so both sides are squares.
Give them the habit: write two lines. First r squared equals 25, then r equals 5. Never jump straight to the radius.
And the sanity check: sketch it. A circle of radius 25 centred near the origin would run off any reasonable grid, which makes the error visible.
Commit first
Commit before you check.
Predict first
A circle has equation (x minus 2) squared plus (y plus 6) squared equals 45. What is its centre?
Correct: (2, negative 6)
Why: The bracket x minus 2 gives h equals positive 2 directly, since the form already subtracts. The bracket y plus 6 is y minus negative 6, so k equals negative 6. Only the bracket containing a plus sign flips. Note that the radius here is root 45, which does not simplify to a whole number — that is perfectly normal and not a sign of an error.
Connect it up
Blank paper.
Draw it
Split the page into two halves. On the left, headed EQUATION, write the standard form and draw arrows from each part to what it gives: h and k with a note that the signs flip, and the right-hand side with a note that it is r squared and needs a root. Underneath, write the completing-the-square recipe: halve the coefficient, square it, add to BOTH sides. On the right, headed ARCS AND SECTORS, draw a circle with a sector marked, and write the fraction theta over 360. Show that fraction multiplying the circumference to give the arc, and multiplying the area to give the sector area. Mark the sector's three edges and write perimeter equals arc plus 2r. At the bottom, write 180 degrees equals pi radians.
Exit ticket
One question before you close the deck.
Predict first
You have completed the square and reached (x minus 4) squared plus (y plus 1) squared equals 49. What do you write next?
Correct: r squared equals 49, so r equals 7
Why: Writing the intermediate line makes the square root a deliberate step rather than one that can be skipped, which is the single most common error on this type. Reporting 49 as the radius is that error. The centre is (4, negative 1) — the third choice flips the wrong bracket. Expanding back is a genuinely useful check, but it comes after extracting the radius rather than instead of it.
Recap
One type, two halves: get to standard form and read, or take a fraction of the whole.
| never do this | do this instead |
|---|---|
| Report the right-hand side as the radius | Write r squared equals it, then take the root |
| Read (x + 5) as a centre at positive 5 | Rewrite it as x minus negative 5 |
| Add completion constants to the left only | Add them to both sides, keeping a running total |
| Give an arc length when asked for a perimeter | Add the two radii |
| Complete the square with a coefficient on x squared | Divide the whole equation through first |
| Convert to degrees when only a fraction is needed | The fraction of a circle is unit-free |
Naruhodo Tutoring — SAT Math question bank export (sat-question-index.json) — and the site's SAT pages to drill this type in isolation, then mixed
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