SAT Math · Geometry and Trigonometry

Circles on the SAT

SAT circle questions use a few tools: the equation (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2, which gives the center (h,k)(h, k) and radius r; arc length and sector area as a fraction of the whole circle; radians; and the fact that a tangent line is perpendicular to the radius. If the equation is expanded, complete the square on x and on y to find the center and radius.

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The key ideas

(x−h)2+(y−k)2=r2⟹center (h,k), radius r(x - h)^2 + (y - k)^2 = r^2 \quad\Longrightarrow\quad \text{center } (h, k), \text{ radius } r

The signs flip: (x+2)2(x + 2)^2 means h=−2h = -2. And the right side is the radius squared, so =49= 49 means a radius of 7.

An arc or a sector is a fraction of the whole circle. That fraction is the central angle over the full turn:

arc length=θ360⋅2πrsector area=θ360⋅πr2\text{arc length} = \frac{\theta}{360} \cdot 2\pi r \qquad \text{sector area} = \frac{\theta}{360} \cdot \pi r^2

In radians the full turn is 2π2\pi, and the formulas become arc length =rθ= r\theta and sector area =12r2θ= \frac{1}{2}r^2\theta.

Worked examples

Example 1: read the equation

Problem What are the center and radius of (x+2)2+(y−5)2=49(x + 2)^2 + (y - 5)^2 = 49?

  1. Rewrite x+2x + 2 as x−(−2)x - (-2), so h=−2h = -2 and k=5k = 5.
  2. The right side is r2r^2.
    r=49=7r = \sqrt{49} = 7

Answer Center (−2,5)(-2, 5), radius 7

Example 2: complete the square

Problem Find the center and radius of x2+y2−8x+6y−11=0x^2 + y^2 - 8x + 6y - 11 = 0.

  1. Group the x terms and the y terms, and move the number to the right.
    (x2−8x)+(y2+6y)=11(x^2 - 8x) + (y^2 + 6y) = 11
  2. Half of −8-8 is −4-4, and (−4)2=16(-4)^2 = 16. Half of 6 is 3, and 32=93^2 = 9. Add both to each side.
    (x2−8x+16)+(y2+6y+9)=11+16+9(x^2 - 8x + 16) + (y^2 + 6y + 9) = 11 + 16 + 9
  3. Factor each group.
    (x−4)2+(y+3)2=36(x - 4)^2 + (y + 3)^2 = 36

Answer Center (4,−3)(4, -3), radius 6

Example 3: arc length and sector area

Problem A circle has radius 9. A central angle of 80∘80^\circ cuts off an arc. Find the arc length and the sector area.

  1. The sector is 80360=29\frac{80}{360} = \frac{2}{9} of the circle. Arc length is that fraction of the circumference 18π18\pi.
    29⋅18π=4π\frac{2}{9} \cdot 18\pi = 4\pi
  2. Sector area is the same fraction of the area 81π81\pi.
    29⋅81π=18π\frac{2}{9} \cdot 81\pi = 18\pi

Answer Arc length 4π4\pi, sector area 18π18\pi

Example 4 (SAT-hard): radians, working backward

Problem A sector of a circle has a central angle of 2π3\frac{2\pi}{3} radians and an area of 24π24\pi. What is the radius?

  1. Use sector area =12r2θ= \frac{1}{2}r^2\theta.
    12r2⋅2π3=24π\frac{1}{2}r^2 \cdot \frac{2\pi}{3} = 24\pi
  2. Simplify the left side.
    πr23=24π\frac{\pi r^2}{3} = 24\pi
  3. Multiply by 3 and divide by π\pi.
    r2=72r^2 = 72
  4. Take the positive root and simplify: 72=36⋅2\sqrt{72} = \sqrt{36 \cdot 2}.

Answer r=62r = 6\sqrt{2}, about 8.49

Common mistakes

  • Reading r2r^2 as r. In (x−1)2+y2=25(x - 1)^2 + y^2 = 25, the radius is 5, not 25. Fix: take the square root of the right side.
  • Keeping the signs inside the parentheses. (x+2)2(x + 2)^2 gives an x-coordinate of −2-2. Fix: the center has the opposite signs.
  • Adding the completed squares to one side only. Adding 16 and 9 on the left means adding them on the right too. Fix: balance every step.
  • Using 180 instead of 360. An 80∘80^\circ arc is 80360\frac{80}{360} of the circle. Fix: in degrees, the whole circle is 360.
  • Mixing degrees and radians. In 12r2θ\frac{1}{2}r^2\theta, θ\theta must be in radians. Fix: convert before you substitute.

Quick method

Practice

5 SAT-style questions

  1. Which equation is a circle with center (3,−1)(3, -1) and radius 4?

    1. (x−3)2+(y+1)2=16(x - 3)^2 + (y + 1)^2 = 16
    2. (x+3)2+(y−1)2=16(x + 3)^2 + (y - 1)^2 = 16
    3. (x−3)2+(y+1)2=4(x - 3)^2 + (y + 1)^2 = 4
    4. (x−3)2+(y−1)2=16(x - 3)^2 + (y - 1)^2 = 16
    Show answer

    Answer: (x−3)2+(y+1)2=16(x - 3)^2 + (y + 1)^2 = 16

    Center (3,−1)(3, -1) gives (x−3)(x - 3) and (y+1)(y + 1), and the right side is 42=164^2 = 16. The second choice flips both signs, the third forgets to square 4, and the last has the wrong y sign.

  2. What is the radius of the circle x2+y2+10x−4y=7x^2 + y^2 + 10x - 4y = 7?

    1. 7\sqrt{7}
    2. 66
    3. 77
    4. 3636
    Show answer

    Answer: 66

    Complete the square: add 25 and 4 to both sides to get (x+5)2+(y−2)2=36(x + 5)^2 + (y - 2)^2 = 36. The radius is 36=6\sqrt{36} = 6. 36 is r2r^2, and 7\sqrt{7} skips completing the square.

  3. A circle has radius 12. What is the length of the arc cut off by a central angle of 45∘45^\circ?

    1. 1.5π1.5\pi
    2. 3π3\pi
    3. 6π6\pi
    4. 18π18\pi
    Show answer

    Answer: 3π3\pi

    45360=18\frac{45}{360} = \frac{1}{8} of the circumference 24π24\pi is 3π3\pi. 18π18\pi is the sector area, and 6π6\pi uses 180 instead of 360.

  4. Which point lies inside the circle (x−1)2+y2=25(x - 1)^2 + y^2 = 25?

    1. (6,0)(6, 0)
    2. (4,3)(4, 3)
    3. (−5,0)(-5, 0)
    4. (1,5)(1, 5)
    Show answer

    Answer: (4,3)(4, 3)

    Put each point in the left side and compare with 25. (4,3)(4, 3): 9+9=18<259 + 9 = 18 < 25, inside. (6,0)(6, 0) and (1,5)(1, 5) give exactly 25, so they are on the circle, and (−5,0)(-5, 0) gives 36, outside.

  5. Student-produced response: a circle centered at the origin passes through the point (5,12)(5, 12). Its area is kπk\pi. What is k?

    Show answer

    Answer: 169

    The radius is the distance from (0,0)(0, 0) to (5,12)(5, 12): 25+144=13\sqrt{25 + 144} = 13. The area is π(13)2=169π\pi (13)^2 = 169\pi, so k=169k = 169.

Frequently asked questions

How do I find the center and radius from an expanded circle equation?

Complete the square. Group the x terms and y terms, move the constant to the right, add half of each linear coefficient squared to both sides, and factor. Then read (h,k)(h, k) and take the square root of the right side for r.

What is the difference between arc length and sector area?

Arc length is the distance along the curved edge, a length. Sector area is the space inside the pie slice, an area. Both are the same fraction of the whole circle: of the circumference for arc length, and of the area for sector area.

What does it mean that a tangent is perpendicular to the radius?

A tangent line touches the circle at one point. The radius drawn to that point meets the tangent at a right angle. SAT questions use that right angle to build a right triangle and apply the Pythagorean theorem.

Sources

  1. College Board: SAT Math, Geometry and Trigonometry skills (circles), accessed October 1, 2026

Try asking Ducky

  • “Why is the center (-2, 5) and not (2, -5)?”
  • “Walk me through completing the square on this circle.”
  • “How do I know if I should use degrees or radians here?”

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