Medium is where most scores are actually won and lost. These questions are not tricky for the sake of it, but every one of them has a wrong answer built to catch a specific shortcut.
Every question below is a real item from the SAT Climb bank, tagged medium by the same difficulty model the app uses to build your practice. Pick an answer before you open the explanation.
Math · Geometry and Trigonometry~1 per testMedium tier
None of these repeat the 25 hard Circles questions, so the two pages together are 45 different questions on this one skill.
Question 1Medium
A circle has center C and circumference 36π. Points P and Q are on the circle. If the length of arc PQ is 9π, what is the measure of central angle PCQ, in degrees?
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Why C is right
The arc is 9π out of total circumference 36π, which is 1/4 of the circle. A full circle is 360∘, so the central angle is 360∘/4=90∘.
Why the others are wrong
AThis incorrectly calculates the fraction of the circle or makes an arithmetic error in the proportion.
BThis correctly finds π/2 radians but the question asks for degrees, not radians.
DThis uses the diameter relationship or incorrectly doubles the correct angle.
Question 2Medium
In the xy-plane, a circle has equation x2+y2+10x−4y+4=0. Point (a, b) is the center of the circle. What is the value of a+b?
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Why A is right
Completing the square: ([MATH]x2+10x+25)[/MATH] + ([MATH]y2−4y+4) = -4 + 25 + 4[/MATH], giving (x+5)2+(y−2)2=25. The center is (-5, 2), so a=−5 and b=2, making a+b=−3.
Why the others are wrong
BThis incorrectly identifies the center as (5, -2) by not properly handling the signs in the standard form, giving 5 + (-2) = 3.
CThis incorrectly uses half the coefficients with wrong signs: -10/2 + (-4)/2 = -5 - 2 = -7.
DThis incorrectly takes the absolute values of half the coefficients: 10/2 + 4/2 = 5 + 2 = 7, or makes a similar computational error.
Question 3Medium
Circle A has equation (x+3)2+(y−5)2=16. Circle B is the image of circle A under a translation 6 units to the left and 2 units down. Which equation represents circle B?
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Why B is right
Circle A has center (-3, 5) and radius 4. A translation 6 units left and 2 units down moves the center to (-3 - 6, 5 - 2) = (-9, 3). The radius remains 4, so the radius squared remains 16, giving equation (x+9)2+(y−3)2=16.
Why the others are wrong
AThis incorrectly translates in the opposite directions, moving right instead of left and down instead of up.
CThis correctly translates the x-coordinate but incorrectly subtracts 2 from the y-coordinate instead of from 5.
DThis treats the translation as a scaling transformation that affects the radius.
Question 4Medium
In a circle with center O, points P, Q, and R lie on the circle. The central angle POQ has a measure of 5π/6 radians. An inscribed angle PRQ intercepts the same arc PQ. What is the measure of angle PRQ, in degrees?
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Why A is right
An inscribed angle is half the central angle that intercepts the same arc. The central angle is 5π/6 radians, which converts to 150 degrees. Therefore, the inscribed angle PRQ measures 150/2 = 75 degrees.
Why the others are wrong
BThis results from incorrectly treating 5π/6 as if it were already in degrees and dividing by some incorrect factor.
CThis results from converting 5π/6 radians to 150 degrees but forgetting to apply the inscribed angle theorem (dividing by 2).
DThis results from doubling the central angle instead of halving it when applying the inscribed angle theorem.
Question 5Medium
A circle has center O and radius 9. A sector of the circle has a central angle measuring 5π/6 radians. What is the area of this sector?
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Why C is right
Sector area is (1/2)r2θ where θ is in radians. With r=9 and θ = 5π/6, area = (1/2)(81)(5π/6)=(81⋅5π)/12=405π/12=135π/4.
Why the others are wrong
AThis uses radius 3 instead of 9, treating it as if reduced by a factor of 3.
BThis omits π from the final calculation.
DThis converts the angle to degrees (150°) but uses that value incorrectly in the area formula.
Question 6Medium
In the xy-plane, a circle has equation x2+y2−8x+6y−11=0. What is the area of the circle?
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Why C is right
To find the area, first determine the radius by rewriting the equation in standard form. Completing the square: ([MATH]x2−8x+16)[/MATH] + ([MATH]y2+6y+9) = 11 + 16 + 9[/MATH], which gives (x−4)2+(y+3)2=36. The radius is 36=6, so the area is π(6)2=36π.
Why the others are wrong
AThis is the radius value, not the area. The area formula requires squaring the radius and multiplying by π.
BThis results from using the diameter (12) as the radius in the area formula. The radius is 6, not 12.
DThis results from incorrectly squaring the value 36 (from r² = 36) to get 1296, then misapplying the constant. The correct area uses r = 6.
Question 7Medium
A circle in the xy-plane has equation x2+y2−6x+4y−12=0. What is the radius of the circle?
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Why A is right
Completing the square for x2 - 6x gives (x−3)2−9, and for y2 + 4y gives (y+2)2−4. The equation becomes (x−3)2+(y+2)2−9−4−12=0, or (x−3)2+(y+2)2=25. The radius is 25=5.
Why the others are wrong
BThis is the diameter, not the radius. The radius is half of 10.
CThis is r², not r. The radius is the square root of 25.
DThis incorrectly treats the constant terms; the correct completion yields r² = 25.
Question 8Medium
Circle A has equation (x−2)2+(y+1)2=9. Circle B is the image of circle A under a translation 3 units to the right and 4 units up. Which equation represents circle B?
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Why A is right
Circle A has center (2, -1) and radius 3. A translation 3 units right and 4 units up moves the center to (2 + 3, -1 + 4) = (5, 3). The radius remains 9, so circle B has equation (x−5)2+(y−3)2=9.
Why the others are wrong
BThis incorrectly subtracts 3 from the x-coordinate and adds 4 to the y-coordinate, reversing the direction of the horizontal translation.
CThis correctly translates the x-coordinate but incorrectly moves the y-coordinate in the wrong direction.
DThis treats the translation as affecting the radius rather than the center coordinates.
Question 9Medium
A circle has center O and radius 9. An inscribed angle intercepts an arc of length 6π. What is the measure of the inscribed angle, in radians?
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Why B is right
Arc length s=rθ gives 6π=9θ, so the central angle θ = 2π/3 radians. An inscribed angle is half the central angle, so the inscribed angle = (2π/3)/2=π/3 radians.
Why the others are wrong
AThis incorrectly applies the half-angle relationship twice or divides by an extra factor.
CThis gives the central angle instead of the inscribed angle, missing the division by 2.
DThis converts π/3 radians to degrees (60°) instead of leaving the answer in radians.
Question 10Medium
A circle has center O and radius 15. A sector of the circle has a central angle measuring 4π/5 radians. What is the area of this sector?
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Why D is right
Sector area is (1/2)r2θ where θ is in radians. With r=15 and θ = 4π/5, area = (1/2)(225)(4π/5)=(225⋅4π)/10=900π/10=90π.
Why the others are wrong
AThis uses an incorrect radius or misapplies the sector area formula.
BThis omits π from the final calculation.
CThis converts the angle to degrees (144°) but uses that value incorrectly in the area formula.
Question 11Medium
A circle has center O and radius 18. Points A and B are on the circle. The measure of central angle AOB is π/3 radians. What is the length of arc AB?
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Why B is right
Arc length is calculated using the formula s=rθ, where r is the radius and θ is the central angle in radians. With r=18 and θ = π/3, the arc length is 18(π/3)=6π.
Why the others are wrong
AThis results from incorrectly using the angle measure π/3 as if it were a fraction to multiply by 18, then omitting the π from the final answer. The correct formula is s = rθ with θ in radians.
CThis results from using the diameter (36) instead of the radius (18) in the arc length formula. The formula s = rθ requires the radius.
DThis results from converting π/3 radians to 60 degrees and treating that as the arc length. Arc length in this context must include π and is calculated as rθ where θ is in radians.
Question 12Medium
A circle in the xy-plane has the equation x2+y2−8x+6y=0. What is the area of the circle?
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Why C is right
Completing the square: x2 - 8x + y2 + 6y=0 becomes (x−4)2−16+(y+3)2−9=0, so (x−4)2+(y+3)2=25. The radius is 5, so the area is πr2 = 25π.
Why the others are wrong
AThis is the square of the radius without π, not the area of the circle.
BThis results from using the diameter (10) instead of the radius (5) in the formula A = πr².
DThis results from squaring the radius incorrectly as (5²)² = 100 instead of 5² = 25.
Question 13Medium
Circle A has equation (x−1)2+y2=36. Circle B is the image of circle A under a translation 5 units to the left and 8 units up. Which equation represents circle B?
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Why C is right
Circle A has center (1, 0) and radius 6. A translation 5 units left and 8 units up moves the center to (1 - 5, 0 + 8) = (-4, 8). The radius remains 6, so circle B has equation (x+4)2+(y−8)2=36.
Why the others are wrong
AThis correctly translates the x-coordinate but incorrectly writes the y-coordinate term as (y + 8)² instead of (y - 8)².
BThis treats the translation as affecting the radius instead of the center.
DThis incorrectly applies the horizontal translation in the opposite direction.
Question 14Medium
A circle has center C. Points P, Q, and R lie on the circle. Point P and point R are endpoints of a diameter. The inscribed angle ∠PQR intercepts an arc that measures 180°. What is the measure of ∠PQR?
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Why C is right
An inscribed angle is half the measure of the central angle that intercepts the same arc. Since the arc is 180° (a semicircle), the inscribed angle measures 180°/2 = 90°. Alternatively, any angle inscribed in a semicircle is a right angle.
Why the others are wrong
AThis incorrectly divides the arc measure by 4 instead of 2.
BThis incorrectly divides the arc measure by 3 instead of 2.
DThis incorrectly uses the arc measure directly without applying the inscribed angle theorem.
Question 15Medium
A circle has center O and radius 9. A sector of the circle has central angle 2π/5 radians. What is the area of this sector?
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Why A is right
The area of a sector is (θ/2π)×πr2, where θ is the central angle in radians. Substituting θ = 2π/5 and r=9 gives (2π/5)/(2π)×π(81)=1/5×81π=81π/5.
Why the others are wrong
BThis incorrectly uses the diameter or makes an error in squaring the radius.
CThis doubles the correct answer, suggesting a misunderstanding of the sector area formula.
DThis results from incorrectly converting radians to degrees before applying the formula.
Question 16Medium
A circle in the xy-plane has equation x2+y2−6x+2y−15=0. What is the diameter of the circle?
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Why B is right
Completing the square for x2 - 6x gives (x−3)2−9, and for y2 + 2y gives (y+1)2−1. The equation becomes (x−3)2+(y+1)2−9−1−15=0, which simplifies to (x−3)2+(y+1)2=25. Since r2 = 25, the radius is 5, and the diameter is 10.
Why the others are wrong
AThis gives the radius instead of the diameter.
CThis incorrectly uses r² directly as the diameter.
DThis incorrectly doubles r² instead of r.
Question 17Medium
A circle in the xy-plane has equation x2+y2−6x+8y=0. What is the area of the circle?
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Why B is right
Completing the square: ([MATH]x2−6x+9)[/MATH] + ([MATH]y2+8y+16) = 9 + 16[/MATH] gives (x−3)2+(y+4)2=25, so the radius is 5. The area is πr2 = 25π.
Why the others are wrong
AThis incorrectly uses the diameter (10) divided by 2 and then squares it, or confuses the radius with another calculation.
CThis omits π from the area formula, calculating only r² = 25.
DThis uses the diameter (10) instead of the radius (5) in the area formula: π(10)² / 4 confusion or double-counting.
Question 18Medium
Circle A has equation (x−8)2+(y+6)2=64. Circle B is the image of circle A under a translation 3 units to the left and 5 units down. Which equation represents circle B?
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Why A is right
Circle A has center (8, -6) and radius 8. A translation 3 units left and 5 units down moves the center to (8 - 3, -6 - 5) = (5, -11). The radius remains 8, so circle B has equation (x−5)2+(y+11)2=64.
Why the others are wrong
BThis incorrectly applies the horizontal translation in the opposite direction.
CThis correctly translates the x-coordinate but incorrectly computes the new y-coordinate as -1 instead of -11.
DThis treats the translation distances as affecting the radius squared.
Question 19Medium
In a circle, an inscribed angle intercepts an arc of length 16π. If the central angle that intercepts the same arc measures 4π/3 radians, what is the measure of the inscribed angle, in radians?
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Why B is right
An inscribed angle is half the central angle intercepting the same arc. The central angle is 4π/3 radians, so the inscribed angle is (4π/3)/2=2π/3 radians.
Why the others are wrong
AThis doubles the central angle instead of halving it, or confuses the relationship direction.
CThis divides 4π/3 by 2 but omits the π, giving 4/3 instead of 2π/3.
DThis converts 2π/3 radians to degrees (120°) but the question asks for radians.
Question 20Medium
A circle has center at the origin and passes through the point (7, 0). What is the area of a sector of this circle with central angle π/4 radians?
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Why A is right
The radius is 7 (distance from origin to (7, 0)). The sector area is (θ/2π)×πr2 = (π/4)/(2π)×π(49)=81×49π=49π/8.
Questions are written by SAT Climb and drawn from its own item bank. SAT® is a registered trademark of College Board, which is not affiliated with and does not endorse SAT Climb.