SAT Math: Circles

Circles: arcs, sectors and equations.

Geometry and Trigonometry · Circles
Both modules
13 practice questions · 4 with free solutions
Calculator allowed (Desmos)

Practise circles →Adaptive SAT module

What the test covers

Arc length and sector area, central angles
Equation of a circle; completing the square

Key ideas

Arc length $= \frac{θ}{360} \times 2 π r$ ; sector area $= \frac{θ}{360} \times π r^{2}$ .
$(x - h)^{2} + (y - k)^{2} = r^{2}$ has centre $(h, k)$ and radius $r$ .
Complete the square in $x$ and in $y$ to find the centre and radius from $x^{2} + y^{2} + D x + E y = F$ .

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.

IB link: IB SL 3.4.

Worked example

What is the radius of the circle with equation $x^{2} + y^{2} + 8 x - 6 y = 11$ in the $x y$ -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. Solutions to the first 3 are open to everyone; the rest need only a free account. Every answer here was checked independently before it was published.

Question 1

What is the radius of the circle with equation $x^{2} + y^{2} - 8 x - 2 y = - 8$ in the $x y$ -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 2

A circle has radius 12. An arc of the circle has central angle $30^{\circ}$ . The length of the arc is $k π$ . What is the value of $k$ ?

Student-produced response: type your answer.

Show the answer and solution

Answer: 2

Arc length $= \frac{30}{360} \times 2 π \times 12 = 2 π$ .

Question 3

A circle has radius 6. An arc of the circle has central angle $120^{\circ}$ . The length of the arc is $k π$ . What is the value of $k$ ?

Student-produced response: type your answer.

Show the answer and solution

Answer: 4

Arc length $= \frac{120}{360} \times 2 π \times 6 = 4 π$ .

Question 4

A circle has radius 9. An arc of the circle has central angle $80^{\circ}$ . The length of the arc is $k π$ . What is the value of $k$ ?

Student-produced response: type your answer.