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Math: Geometry and Trigonometry, 3 original questions

SAT Circles Practice: Questions and Answers

Read circle centers and radii, calculate arc length and complete the square in three original SAT-style practice questions with explanations for all choices.

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The short answer

Circle questions connect radius, angle and coordinate geometry. For an equation in the form (x − h)² + (y − k)² = r², read the center as (h, k) and take the positive square root of the right side for the radius. For arcs and sectors, use the central angle's fraction of a full circle. If an equation is expanded, complete the square separately in x and y. Keep degrees and radians distinct, and check whether the question requests a radius, arc length or area.

Method at a glance

  1. 1Mark the center and radius
  2. 2Identify the angular fraction
  3. 3Use circumference or area
  4. 4Complete squares when needed

Try the 3 questionsRead the full method

Step-by-step method

Use these steps in order, then apply them to the questions below.

  1. Step 1: Mark the center and radius

    Read the signs inside squared parentheses carefully. A radius is a length, so r is the positive square root of r².

  2. Step 2: Identify the angular fraction

    A full circle is 360 degrees or 2π radians. Divide a central angle by the matching full-circle measure.

  3. Step 3: Use circumference or area

    Multiply the angular fraction by 2πr for an arc length or by πr² for a sector area.

  4. Step 4: Complete squares when needed

    Group x and y terms, add the square-completing constants to both sides, then read the resulting center and radius.

Try 3 SAT-style questions

These are original SAT-style questions written by PeakSAT, not official College Board questions. Choose an answer and check it to see why each choice is right or wrong, or open the worked answer.

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Question 1 of 3

A circle has equation (x − 2)² + (y + 3)² = 49. Which choice gives its center?

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Correct answer

Choice A: (2, −3)

Explanation

Compare with (x − h)² + (y − k)² = r². Here h = 2 and y + 3 is y − (−3), so k = −3. The center is (2, −3). The radius is 7, but it is not part of the requested center.

Why each choice is right or wrong

  1. Choice A

    Correct answer

    Correct: both squared terms are zero at (2, −3).

  2. Choice B

    Both coordinate signs are reversed.

  3. Choice C

    The y-coordinate must be −3 because the term is y + 3.

  4. Choice D

    The x-coordinate must be positive 2 because the term is x − 2.

Question 2 of 3

A circle has radius 12 units. What is the length of an arc with central angle 60 degrees?

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Correct answer

Choice B: 4π units

Explanation

The arc is 60/360 = 1/6 of the circle. The circumference is 2π × 12 = 24π units. One sixth of 24π is 4π units, the requested arc length.

Why each choice is right or wrong

  1. Choice A

    This omits the factor of 2 in the circumference before taking one sixth.

  2. Choice B

    Correct answer

    Correct: (60/360) × 24π = 4π.

  3. Choice C

    This does not apply the arc's one-sixth fraction correctly.

  4. Choice D

    This is the full circumference, not the 60-degree arc.

Question 3 of 3

What is the radius of the circle x² + y² − 8x + 6y = 0?

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Correct answer

Choice A: 5

Explanation

Group terms and complete squares: x² − 8x + 16 = (x − 4)², and y² + 6y + 9 = (y + 3)². Add 16 + 9 to both sides to get (x − 4)² + (y + 3)² = 25. Thus r² = 25 and r = 5.

Why each choice is right or wrong

  1. Choice A

    Correct answer

    Correct: the completed equation has radius squared 25, so the radius is 5.

  2. Choice B

    Adding the magnitudes of center coordinates does not give the radius.

  3. Choice C

    25 is r², not r.

  4. Choice D

    The square-completing constants add to 25, not 7.

Common mistakes

  • Copying the signs inside parentheses directly into the center. In x − h, the center coordinate is h.

  • Reading r² as the radius. Take its positive square root before using a length formula.

  • Using circle area to find an arc length. An arc is a fraction of the circumference, measured in length units.

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Common questions

How can I recognize a circle in coordinate form?

Its standard form is (x − h)² + (y − k)² = r² with a positive r². The center is (h, k).

What fraction do I use for a central angle in radians?

Divide the angle by 2π. For degrees, divide by 360. Use the same unit in the numerator and denominator.

How is a sector different from an arc?

A sector is a portion of the circle's interior and has area. An arc is a portion of its boundary and has length.

Sources

  1. College Board: Geometry and Trigonometrysatsuite.collegeboard.org
  2. College Board: Student Question Bank Math skillssatsuite.collegeboard.org

Sources are linked for reference; they do not endorse PeakSAT. How we prepare our guides