Circles — arcs, sectors, angles, equations — ACT practice questions

On the ACT Mathematics test, Circles — arcs, sectors, angles, equations covers radius, circumference, area, arc measure, sector area, central and inscribed angles, and the standard circle equation in the coordinate plane. A student has to convert between circumference and area, compute the area of a sector from a radius and a central angle, and write an equation from a given center and radius. Items are multiple choice, with numeric values or algebraic equations as options. Common traps include swapping diameter for radius, mixing arc length with sector area, and flipping the signs of the center coordinates in the equation.

Section: Mathematics 45 questions on the paper
58 practice questions in the app

Sample questions

Question 1

In the standard $(x,y)$ coordinate plane, a circle has center $(-3,4)$ and radius 5. Which of the following is an equation of the circle?

A $(x-3)^2+(y+4)^2=25$
B $(x+3)^2+(y-4)^2=5$
C $(x-3)^2+(y+4)^2=5$
D $(x+3)^2+(y-4)^2=25$

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Answer: D — $(x+3)^2+(y-4)^2=25$

A circle with center $(h,k)$ and radius $r$ has equation $(x-h)^2+(y-k)^2=r^2$ . With $h=-3$ , $k=4$ , and $r=5$ , this is $(x+3)^2+(y-4)^2=25$ .

Question 2

A sector of a circle with radius 6 has central angle $60^\circ$ . What is the area of the sector?

A $2\pi$
B $6\pi$
C $12\pi$
D $36\pi$

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Answer: B — $6\pi$

Sector area $=\dfrac{60}{360}\pi r^2=\dfrac{1}{6}\pi(36)=6\pi$ .

Question 3

A circle has a circumference of $10\pi$ units. What is the area of the circle, in square units?

A $5\pi$
B $10\pi$
C $20\pi$
D $25\pi$

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Answer: D — $25\pi$

From $2\pi r=10\pi$ , the radius is 5. Therefore, the area is $\pi r^2=25\pi$ square units.

Practice 58 Circles — arcs, sectors, angles, equations questions in the app

Circles — arcs, sectors, angles, equations — ACT practice questions

Sample questions

More ACT topics

Mathematics