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ACT Math : Sectors

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Example Question #1 : How To Find The Length Of An Arc

What is the area of the sector of a circle with a central angle of 120 degrees and a radius of 5cm ? Simplify any fractions and leave your answer in terms of $\pi$ .

Possible Answers:

$\frac{3\pi}{25}cm^2$

$25\pi \cm^2$

$\frac{25\pi}{3} cm^2$

$3000\pi cm^2$

$24\pi cm^2$

Correct answer:

$\frac{25\pi}{3} cm^2$

Explanation:

The formula for the area of a sector of a circle is:

$\frac{central\: angle}{360}*\pi r^2$

The central angle given is 120 thus:

$\frac{120}{360}*\pi (5cm)^2= \frac{1}{3}\pi 25cm^2 = \frac{25\pi}{3}cm^2$

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Example Question #2 : How To Find The Length Of An Arc

A water wheel turns a $120^{\circ}$ arc every minute. If the radius of the wheel is , how far in meters does the wheel turn along its edge each minute?

Possible Answers:

$8\pi m$

$4\pi m$

$10\pi m$

$\frac{\pi}{3} m$

$\frac{5\pi}{3} m$

Correct answer:

$4\pi m$

Explanation:

If the radius is , then the circumference of the wheel is:

$C = 2r\pi = 12\pi$

If the wheel turns $120^{\circ}$ each minute, then it turns $\frac{120^{\circ}}{360^{\circ}} = \frac{1}{3}$ of the circumference each minute.

$d = 12\pi \cdot \frac{1}{3} = 4\pi$

Thus, the wheel turns $4\pi m$ each minute.

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Example Question #3 : How To Find The Length Of An Arc

What is the length of the arc ABC ?

arc2

The total area of the circle is $64\pi \, in^{2}$ and the area of the shaded region is $12\pi \,in^{2}$ .

Possible Answers:

$16\pi\:in$

$\frac{12\pi}{5}\:in$

$\frac{3\pi}{16}\:in$

$3\pi\:in$

$8\:in$

Correct answer:

$3\pi\:in$

Explanation:

If the area of the circle is $64\pi\:in^2$ , the radius can be found using the formula for the area of a circle:
$A=\pi r^2$

For our data, this is:

$64\pi=\pi r^2$

r^2=64

Therefore, $r=8\:in$

Now, the circumference of the circle is defined as:

$C=2\pi r$

For our data, this is:

$C=2*8*\pi=16\pi\:in$

Now, we know that a sector is a percentage of the total area. This percentage is easily calculated:

$\frac{12\pi\:in^2}{64\pi\:in^2}=\frac{3}{16}$

So, the length of the arc will merely be the same percentage, but now applied to the circumference:

$\frac{3}{16} * 16\pi\:in = 3\pi\:in$

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Example Question #1 : How To Find The Angle Of A Sector

Circle

In the circle above, the length of arc BC is 100 degrees, and the segment AC is a diameter. What is the measure of angle ADB in degrees?

Possible Answers:

cannot be determined

100

Correct answer:

Explanation:

Since we know that segment AC is a diameter, this means that the length of the arc ABC must be 180 degrees. This means that the length of the arc AB must be 80 degrees.

Since angle ADB is an inscribed angle, its measure is equal to half of the measure of the angle of the arc that it intercepts. This means that the measure of the angle is half of 80 degrees, or 40 degrees.

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Example Question #2 : How To Find The Angle Of A Sector

What is the angle of a sector of area in^2 on a circle having a radius of $15\:in$ ?

Possible Answers:

$15.22^{\circ}$

$0.06^{\circ}$

$22.92^{\circ}$

$3.00^{\circ}$

$72.00^{\circ}$

Correct answer:

$22.92^{\circ}$

Explanation:

To begin, you should compute the complete area of the circle:

$A=\pi r^2$

For your data, this is:

$A=15^2\pi=225\pi$

Now, to find the angle measure of a sector, you find what portion of the circle the sector is. Here, it is:

$\frac{45}{225\pi}$

Now, multiply this by the total 360 degrees in a circle:

$\frac{45}{225\pi}*360=22.918311805212$

Rounded, this is $22.92^{\circ}$ .

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Example Question #3 : How To Find The Angle Of A Sector

What is the angle of a sector that has an arc length of 13.5 on a circle of diameter ?

Possible Answers:

$35.81^{\circ}$

$14.24^{\circ}$

$10.74^{\circ}$

$194.14^{\circ}$

$128.92^{\circ}$

Correct answer:

$128.92^{\circ}$

Explanation:

The first thing to do for this problem is to compute the total circumference of the circle. Notice that you were given the diameter. The proper equation is therefore:

$C=\pi d$

For your data, this means,

$C=12\pi$

Now, to compute the angle, note that you have a percentage of the total circumference, based upon your arc length:

$\frac{13.5}{12\pi}*360=128.9155039044336$

Rounded to the nearest hundredth, this is $128.92^{\circ}$ .

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ACT Math : Sectors

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #1 : How To Find The Length Of An Arc

Example Question #2 : How To Find The Length Of An Arc

Example Question #3 : How To Find The Length Of An Arc

Example Question #1 : How To Find The Angle Of A Sector

Example Question #2 : How To Find The Angle Of A Sector

Example Question #3 : How To Find The Angle Of A Sector

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