How to find the length of an arc - ACT Math

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Question

In the circle above, the angle A in radians is $\frac{\pi}{4}$

What is the length of arc A?

Answer

Circumference of a Circle = $2\pi r$

Arc Length

$=\frac{Angle A}{2\pi}\ast2\pi r$

$=\frac{\frac{\pi}{4}}{2\pi}\ast2\pi r$

$=\frac{\pi}{(4)2\pi}\ast2\pi r$

$=\frac{1}{8}\ast2\pi r=\frac{\pi r}{4}$

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Question

If a circle has a circumference of , what is the measure of the arc contained by a degree angle located at the center of the circle?

Answer

A circle has a total of degrees. If our angle is located at the center and is degrees, we can do $\frac{60}{360}=\frac{1}{6}$ to see that our angle makes up $\frac{1}{6}$ of the complete circle.

Therefore, our arc is going to be $\frac{1}{6}$ of our total circumference. $24\cdot \frac{1}{6}=\frac{24}{6}=4$

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Question

The figure above is a circle with center at and a radius of . This figure is not drawn to scale.

What is the length of the arc in the figure above?

Answer

Recall that the length of an arc is merely a percentage of the circumference. The circumference is found by the equation:

$C=2\pi r$

For our data, this is:

$C=2\pi * 15:cm=30\pi:cm$

Now the percentage for our arc is based on the angle $75^{\circ}$ and the total degrees in a circle, namely, $360^{\circ}$ .

So, the length of the arc is:

$\frac{75^{\circ}}{360^{\circ}} * 30\pi:cm=\frac{75\pi}{12}:cm=\frac{25\pi}{4}:cm$

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Question

What is the length of the arc ?

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

Answer

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:

$64\pi=\pi r^2$

Therefore,

Now, the circumference of the circle is defined as:

$C=2\pi r$

For our data, this is:

$C=28\pi=16\pi:in$

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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Question

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

Answer

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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Question

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?

Answer

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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