The rules that solve most questions
- Circumference = 2πr; area = πr².
- Arc length = (central angle / 360°) × 2πr; sector area = (central angle / 360°) × πr².
- Equation: (x − h)² + (y − k)² = r² has center (h, k) and radius r.
- If the equation isn't in that form, complete the square for x and for y.
Traps the wrong answers are built on
- Reading the center as (−h, −k) from (x − h)² + (y − k)².
- Answering r² as the radius.
Worked example
The equation of a circle in the xy-plane is (x − 5)² + (y + 2)² = 64. What are the center and radius of the circle?
- A. (−5, 2) and 8
- B. (−5, 2) and 64
- C. (5, −2) and 64
- D. (5, −2) and 8
Answer: D. The standard form is (x − h)² + (y − k)² = r², with center (h, k) and radius r. Here h = 5 and k = −2 (watch the signs: x − 5 means h = 5). r² = 64, so r = 8.
Practice set
Answer each question to see whether you're right and why. When you finish, draw a new set: the numbers change every time.
0 of 10 answered · 0 correct · set 1
- Question 1Medium
The equation of a circle in the xy-plane is (x − 5)² + (y + 2)² = 64. What are the center and radius of the circle?
Show answer and explanation
Answer: D
The standard form is (x − h)² + (y − k)² = r², with center (h, k) and radius r.
Here h = 5 and k = −2 (watch the signs: x − 5 means h = 5).
r² = 64, so r = 8.
- Question 2Medium
The equation of a circle in the xy-plane is (x − 8)² + (y + 3)² = 49. What are the center and radius of the circle?
Show answer and explanation
Answer: D
The standard form is (x − h)² + (y − k)² = r², with center (h, k) and radius r.
Here h = 8 and k = −3 (watch the signs: x − 8 means h = 8).
r² = 49, so r = 7.
- Question 3Medium
The equation of a circle in the xy-plane is (x + 7)² + (y − 1)² = 36. What are the center and radius of the circle?
Show answer and explanation
Answer: A
The standard form is (x − h)² + (y − k)² = r², with center (h, k) and radius r.
Here h = −7 and k = 1 (watch the signs: x + 7 means h = −7).
r² = 36, so r = 6.
- Question 4Medium
A circle has a circumference of 18π. What is the area of the circle?
Show answer and explanation
Answer: A
Circumference = 2πr, so 2πr = 18π and r = 9.
Area = πr² = π(9)² = 81π.
- Question 5Medium
A circle has a circumference of 22π. What is the area of the circle?
Show answer and explanation
Answer: C
Circumference = 2πr, so 2πr = 22π and r = 11.
Area = πr² = π(11)² = 121π.
- Question 6Medium
A circle has a circumference of 14π. What is the area of the circle?
Show answer and explanation
Answer: B
Circumference = 2πr, so 2πr = 14π and r = 7.
Area = πr² = π(7)² = 49π.
- Question 7Harder
A circle has a radius of 5. What is the length of an arc formed by a central angle of 240°?
Show answer and explanation
Answer: B
The full circumference is 2π(5) = 10π.
A 240° angle takes 240/360 of the circle.
Arc length = 240/360 × 10π = 20/3π.
- Question 8Harder
A circle has a radius of 5. What is the length of an arc formed by a central angle of 60°?
Show answer and explanation
Answer: A
The full circumference is 2π(5) = 10π.
A 60° angle takes 60/360 of the circle.
Arc length = 60/360 × 10π = 5/3π.
- Question 9Medium
The equation of a circle in the xy-plane is (x − 7)² + (y − 6)² = 9. What are the center and radius of the circle?
Show answer and explanation
Answer: C
The standard form is (x − h)² + (y − k)² = r², with center (h, k) and radius r.
Here h = 7 and k = 6 (watch the signs: x − 7 means h = 7).
r² = 9, so r = 3.
- Question 10Harder
A circle has a radius of 8. What is the length of an arc formed by a central angle of 150°?
Show answer and explanation
Answer: B
The full circumference is 2π(8) = 16π.
A 150° angle takes 150/360 of the circle.
Arc length = 150/360 × 16π = 20/3π.
More Geometry and Trigonometry practice
Questions people ask
Is this circles practice like the real SAT?
The questions follow the format of the digital SAT's Geometry and Trigonometry domain: four answer choices built from common mistakes. They are written for this site; they are not official College Board questions.
How many practice questions are there?
As many as you want. Each math question is generated from a template with new numbers, and its answer is checked by code. Use "New set" for another ten.
Where is the domain on the test?
Geometry and Trigonometry is one of the four content domains of the SAT Math section, alongside Algebra, Advanced Math, Problem-Solving and Data Analysis.