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ACT Math : Circles

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Example Questions

Example Question #1 : How To Find The Length Of A Chord

In the figure below, and are radii of the circle. If the radius of the circle is 8 units, how long, in units, is chord ?

Circle

Possible Answers:

$8\sqrt2$

$4\sqrt2$

$16\sqrt2$

Correct answer:

$8\sqrt2$

Explanation:

The radii and the chord form right triangle AOB . Use the Pythagorean Theorem to find the length of .

$8^2+8^2={AB}^2$

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Example Question #1 : Circles

A circle has a maximum chord length of inches. What is its radius in inches?

Possible Answers:

Correct answer:

Explanation:

Keep in mind that the largest possible chord of a circle is its diameter.

This means that the diameter of the circle is inches, which means an -inch radius.

Remember that d = 2r ,

$\\16=2r\\ \\ \frac{16}{2}=\frac{2r}{2}\\ \\ 8=r$

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Example Question #1 : Circles

In the circle with center A, shown below, the length of radius $\overline{AB}$ is 4 mm. What is the length of chord $\overline{BC}$ ?

Circle_chord

Possible Answers:

$2\sqrt{3}$

$4\sqrt{3}$

$8\sqrt{2}$

$2\sqrt{2}$

$4\sqrt{2}$

Correct answer:

$4\sqrt{2}$

Explanation:

In order to solve this question, we need to notice that the radii of the circle and chord $\overline{BC}$ form a right triangle with $\overline{BC}$ as the hypotenuse. We can use the Pythagorean theorem to find chord $\overline{BC}$ .

a^2 + b^2 = c^2

$4^2 + 4^2 = \overline{BC}^2$

$32 = \overline{BC}^2$

$\sqrt{32} = \overline{BC}$

$\sqrt{16\cdot 2} = \overline{BC}$

$4\sqrt{2} = \overline{BC}$

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Example Question #1 : How To Find The Distance Between Clock Hands

A church commissioned a glassmaker to make a circular stained glass window that was two feet in diameter. What is the area of the stained glass window, to the nearest square foot?

Possible Answers:

$12\:ft^2$

$5\:ft^2$

$3\:ft^2$

$8\:ft^2$

$2\:ft^2$

Correct answer:

$3\:ft^2$

Explanation:

To answer this question, we need to find the area of a circle.

To find the area of a circle, we will use the following equation:

$area=\pi r^{2}$ , where is the radius.

We are given the diameter of the circle, which is $2\:ft$ . To find the radius of a circle, we divide the diameter by two. So, for this data:

$radius=\frac{diameter}{2}=\frac{2\:ft}{2}=1\:ft$

So, our radius is $1\:ft$ .

We can now plug our radius into our equation for the area of a circle:

$area=\pi r^{2}=\pi \cdot (1\:ft)^{2}=\pi \cdot 1\:ft^2=\pi\:ft^2=3.14\:ft^2$

The answer asks us what the area is to the nearest square foot. Therefore, we must round our answer to the nearest whole number. To round, we will round up if the digit right before your desired place value is 5, 6, 7, 8, or 9, and we will round down if it is 0, 1, 2, 3, or 4. Because we want the nearest square foot, we will look at the tenths place to aid in our rounding. Because there is a 1 in the tenths place, we will round down. So:

$3.14\:ft^2\rightarrow3\:ft^2$

Therefore, our answer is $3\:ft^2$ .

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Example Question #1 : How To Find The Angle Of Clock Hands

What is the measure of the larger angle formed by the hands of a clock at 5:00 ?

Possible Answers:

$150^{\circ}$

$260^{\circ}$

$180^{\circ}$

$210^{\circ}$

$240^{\circ}$

Correct answer:

$210^{\circ}$

Explanation:

Like any circle, a clock contains a total of $360^{\circ}$ . Because the clock face is divided into equal parts, we can find the number of degrees between each number by doing $\frac{360}{12}=30^{\circ}$ . At 5:00 the hour hand will be at 5 and the minute hand will be at 12. Using what we just figured out, we can see that there is an angle of $150^{\circ}$ between the two hands. We are looking for the larger angle, however, so we must now do $360^{\circ}-150^{\circ}=210^{\circ}$ .

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Example Question #1 : How To Find The Distance Between Clock Hands

Assuming the clock hands extend all the way to the edge of the clock, what is the distance between the clock hands of an analog clock with a radius of 6 inches if the time is 2:30? Reduce any fractions in your answer and leave your answer in terms of $\pi$

Possible Answers:

$\frac{17}{2}\pi$ inches

$14\pi$ inches

$7\pi$ inches

$\frac{14}{3}\pi$ inches

$\frac{7}{2}\pi$ inches

Correct answer:

$\frac{7}{2}\pi$ inches

Explanation:

To find the distance between the clock hands, first find the angle between the clock hands and use that as your central angle to find the porportion of the circumference.
The angle between the hands is $105^{\circ}$ :

$\\ \frac{105}{360} * 2*\pi*r \\ \\ \frac{105}{360}*2\pi * 6 \\ \\= \frac{7}{2}\pi inches$

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Example Question #2 : Circles

An watch with a face radius of $35\textup{cm}$ displays the current time as $7\textup{:}20$ . Assuming each hand reaches to the edge of the clock face, what is the distance, in $\textup{cm}$ , of the minor arc between the hour and minute hand? Leave $\pi$ in your answer. Assume a $\textup{12-hour}$ display (not a military clock).

Possible Answers:

$\frac{4\pi}{17}\textup{cm}$

$\frac{130\pi}{7}\textup{cm}$

$\frac{175\pi}{9}\textup{cm}$

$\frac{15\pi}{19}\textup{cm}$

$\frac{27\pi}{7}\textup{cm}$

Correct answer:

$\frac{175\pi}{9}\textup{cm}$

Explanation:

To start, we must calculate the circumference along the clock face. Since we are required to leave $\pi$ in our answer, this is as easy as following our equation for circumference:

$C=2r\pi = 70\pi$

Next, we must figure out the positions of the hands. At $7\textup{:}20$ , the clock's minute hand would be exactly on the , and the hour hand would be $\frac{20}{60} = \frac{1}{3}$ of the way between the and the . Therefore, the total distance d(m) in terms of hour markings is $d(m) = 7\frac{1}{3} - 4 = 3\frac{1}{3}$ .

To find the distance between adjacent numbers, divide the circumference by :

$d(h) = \frac{C}{12} = \frac{70\pi}{12} = \frac{35\pi}{6}$ , where d(h) is the distance in inches between adjacent hours on the clock face.

Finally, multiply this distance by the number of hours between the hands.

$d = d(m)\cdot d(h) = 3\frac{1}{3}\cdot\frac{35\pi}{6} = \frac{10}{3}\cdot \frac{35\pi}{6} = \frac{175\pi}{9}$ .

Thus, the hands are $\frac{175\pi}{9}\textup{cm}$ apart.

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Example Question #5 : Circles

An military ( $\textup{24-hour}$ ) clock with a face diameter of $3\textup{ft}$ displays the current time as $13\textup{:}30$ . Assuming each hand reaches to the edge of the clock face, what is the distance, in feet, of the minor arc between the hour and minute hand? Leave $\pi$ in your answer.

Possible Answers:

$\frac{5\pi}{4}\textup{ft}$

$\frac{3\pi}{16}\textup{ft}$

$\frac{5\pi}{16}\textup{ft}$

$\frac{\pi}{2}\textup{ft}$

$\frac{3\pi}{4}\textup{ft}$

Correct answer:

$\frac{3\pi}{16}\textup{ft}$

Explanation:

To start, we must calculate the circumference along the clock face. Since we are required to leave $\pi$ in our answer, this is as easy as following our equation for circumference:

$C=D\pi = 3\pi$

Next, we must figure out the positions of the hands. For this problem, the clock has 24 evenly spaced numbers instead of 12, so we must cut the distance between each number mark in half compared to a normal clock face. At 13:30 , the clock's minute hand would be exactly on the , and the hour hand would be $\frac{30}{60} = \frac{1}{2}$ of the way between the and the (remember, there are still only seconds in a minute even on a 24-hour clock. Therefore, the total distance d(m) in terms of hour markings is $d(m) = 13\frac{1}{2} -12 = 1\frac{1}{2}$ .

To find the distance between adjacent numbers, divide the circumference by :

$d(h) = \frac{C}{24} = \frac{3\pi}{24} = \frac{1\pi}{8}$ , where d(h) is the distance in feet between adjacent hours on the clock face.

Finally, multiply this distance by the number of hours between the hands.

$d = d(m)\cdot d(h) = 1\frac{1}{2}\cdot\frac{\pi}{8} = \frac{3\pi}{16}$ .

Thus, the hands are $\frac{3\pi}{16}$ feet apart.

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Example Question #4 : How To Find The Distance Between Clock Hands

An outdoor clock with a face diameter of $18\textup{ft}$ displays the current time as $05\textup{:}45$ . Assuming each hand reaches to the edge of the clock face, what is the distance, in feet, of the minor arc between the hour and minute hand? Leave $\pi$ in your answer. Assume a $\textup{12-hour}$ display (not a military clock).

Possible Answers:

$\frac{13\pi}{3}\textup{ft}$

$4\pi\textup{ft}$

$\frac{39\pi}{8}\textup{ft}$

$\frac{3\pi}{4}\textup{ft}$

$\frac{33\pi}{2}\textup{ft}$

Correct answer:

$\frac{39\pi}{8}\textup{ft}$

Explanation:

To start, we must calculate the circumference along the clock face. Since we are required to leave $\pi$ in our answer, this is as easy as following our equation for circumference:

$C=D\pi = 18\pi$

Next, we must figure out the positions of the hands. At $05\textup{:}45$ , the clock's minute hand would be exactly on the , and the hour hand would be $\frac{45}{60} = \frac{3}{4}$ of the way between the and the . Therefore, the total distance d(m) in terms of hour markings is $d(m) = 9- 5\frac{3}{4} = 3\frac{1}{4}$ .

To find the distance between adjacent numbers, divide the circumference by :

$d(h) = \frac{C}{12} = \frac{18\pi}{12} = \frac{3\pi}{2}$ , where d(h) is the distance in inches between adjacent hours on the clock face.

Finally, multiply this distance by the number of hours between the hands.

$d = d(m)\cdot d(h) = 3\frac{1}{4}\cdot\frac{3\pi}{2} = \frac{39\pi}{8}$ .

Thus, the hands are $\frac{39\pi}{8}$ feet apart.

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Example Question #1 : How To Find The Distance Between Clock Hands

Find the distance between the hour and minute hand of a clock at 2:00 with a hand length of .

Possible Answers:

$4\pi$

$12\pi$

$9\pi$

$2\pi$

Correct answer:

$2\pi$

Explanation:

To find the distance, first you need to find circumference. Thus,

$C=2\pi{r}=2\cdot\pi\cdot6=12\pi$

Then, multiply the fraction of the clock they cover. Thus,

$12\pi\cdot\frac{2}{12}=2\pi$

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ACT Math : Circles

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #1 : How To Find The Length Of A Chord

Example Question #1 : Circles

Example Question #1 : Circles

Example Question #1 : How To Find The Distance Between Clock Hands

Example Question #1 : How To Find The Angle Of Clock Hands

Example Question #1 : How To Find The Distance Between Clock Hands

Example Question #2 : Circles

Example Question #5 : Circles

Example Question #4 : How To Find The Distance Between Clock Hands

Example Question #1 : How To Find The Distance Between Clock Hands

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