SAT Math: Circles
Circles: arcs, sectors and equations.
- Geometry and Trigonometry · Circles
- Both modules
- 13 practice questions
- Calculator allowed (Desmos)
What the test covers
- Arc length and sector area, central angles
- Equation of a circle; completing the square
Key ideas
- Arc length \(=\tfrac{\theta}{360}\times2\pi r\); sector area \(=\tfrac{\theta}{360}\times\pi r^2\).
- \((x-h)^2+(y-k)^2=r^2\) has centre \((h,k)\) and radius \(r\).
- Complete the square in \(x\) and in \(y\) to find the centre and radius from \(x^2+y^2+Dx+Ey=F\).
Common mistakes
- The right-hand side is \(r^2\): take the square root for the radius.
- Signs: \((x+4)^2\) means the centre has \(x=-4\).
Do it in Desmos
Type the circle equation into Desmos and click the centre and edge points.
Worked example
Worked example
What is the radius of the circle with equation \(x^2+y^2+8x-6y=11\) in the \(xy\)-plane?
- \(6\)
- \(\sqrt{11}\)
- \(5\)
- \(36\)
Answer: A: \(6\)
Complete the squares: \((x+4)^2-16+(y-3)^2-9=11\), so \((x+4)^2+(y-3)^2=36\). The radius is 6.
Practice questions
Try each question before opening the solution. Desmos or your own calculator is allowed on every SAT Math question. Every answer here was re-checked independently by computer before it was published.
Question 1
A circle has radius 9. An arc of the circle has central angle \(80^\circ\). The length of the arc is \(k\pi\). What is the value of \(k\)?
Show the answer and solution
Answer: 4
Arc length \(=\tfrac{80}{360}\times2\pi\times9=4\pi\), so \(k=4\).
Question 2
A circle in the \(xy\)-plane has center \((2,-1)\) and passes through the point \((5,3)\). Which is an equation of the circle?
- \((x-2)^2+(y+1)^2=5\)
- \((x-2)^2+(y+1)^2=25\)
- \((x+2)^2+(y-1)^2=25\)
- \((x-5)^2+(y-3)^2=25\)
Show the answer and solution
Answer: B: \((x-2)^2+(y+1)^2=25\)
The radius is the distance from \((2,-1)\) to \((5,3)\): \(\sqrt{9+16}=5\). So \((x-2)^2+(y+1)^2=25\).
Question 3
What is the radius of the circle with equation \(x^2+y^2+10x-8y=8\) in the \(xy\)-plane?
- \(9\)
- \(49\)
- \(2 \sqrt{2}\)
- \(7\)
Show the answer and solution
Answer: D: \(7\)
Complete the squares: \((x+5)^2+(y-4)^2=8+25+16=49\). The radius is \(7\).
Question 4
What is the radius of the circle with equation \(x^2+y^2-8x-2y=-8\) in the \(xy\)-plane?
- \(9\)
- \(5\)
- \(3\)
- \(2 \sqrt{2}\)
Show the answer and solution
Answer: C: \(3\)
Complete the squares: \((x-4)^2+(y-1)^2=-8+16+1=9\). The radius is \(3\).
Question 5
What is the radius of the circle with equation \(x^2+y^2+6x-6y=-9\) in the \(xy\)-plane?
- \(3\)
- \(5\)
- \(4\)
- \(9\)
Show the answer and solution
Answer: A: \(3\)
Complete the squares: \((x+3)^2+(y-3)^2=-9+9+9=9\). The radius is \(3\).
Question 6
What is the radius of the circle with equation \(x^2+y^2-2x-2y=7\) in the \(xy\)-plane?
- \(5\)
- \(9\)
- \(3\)
- \(\sqrt{7}\)
Show the answer and solution
Answer: C: \(3\)
Complete the squares: \((x-1)^2+(y-1)^2=7+1+1=9\). The radius is \(3\).
Question 7
What is the radius of the circle with equation \(x^2+y^2-8x+10y=23\) in the \(xy\)-plane?
- \(64\)
- \(8\)
- \(\sqrt{23}\)
- \(10\)
Show the answer and solution
Answer: B: \(8\)
Complete the squares: \((x-4)^2+(y+5)^2=23+16+25=64\). The radius is \(8\).
Question 8
A circle has radius 12. An arc of the circle has central angle \(30^\circ\). The length of the arc is \(k\pi\). What is the value of \(k\)?
Show the answer and solution
Answer: 2
Arc length \(=\tfrac{30}{360}\times2\pi\times12=2\pi\).
Question 9
A circle has radius 8. An arc of the circle has central angle \(45^\circ\). The length of the arc is \(k\pi\). What is the value of \(k\)?
Show the answer and solution
Answer: 2
Arc length \(=\tfrac{45}{360}\times2\pi\times8=2\pi\).
Question 10
A circle has radius 18. An arc of the circle has central angle \(40^\circ\). The length of the arc is \(k\pi\). What is the value of \(k\)?
Show the answer and solution
Answer: 4
Arc length \(=\tfrac{40}{360}\times2\pi\times18=4\pi\).
Question 11
A circle has radius 10. An arc of the circle has central angle \(36^\circ\). The length of the arc is \(k\pi\). What is the value of \(k\)?
Show the answer and solution
Answer: 2
Arc length \(=\tfrac{36}{360}\times2\pi\times10=2\pi\).
Question 12
A circle has radius 6. An arc of the circle has central angle \(120^\circ\). The length of the arc is \(k\pi\). What is the value of \(k\)?
Show the answer and solution
Answer: 4
Arc length \(=\tfrac{120}{360}\times2\pi\times6=4\pi\).
Keep going
- Previous topic: Right triangles and trigonometry
- Next topic: Linear equations in one variable
- All SAT Math skills · Adaptive SAT Math module simulator · Official SAT practice (Bluebook, Khan Academy)
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