How to find the length of an arc - Math

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Question

The clock at the town square has a minute hand eight feet long. How far has its tip traveled since noon if it is now 12:58 PM?

Answer

This question is asking for the length of an arc corresponding to $\frac{58}{60}$ of a circle with radius eight feet. The question can be answered by evaluating for :

$$\frac{58}{60}$ \cdot 2 \pi r=\frac{58}{60}$ \cdot 2 \pi \cdot 8 \approx 48.6$

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Question

Note: Figure NOT drawn to scale

Refer to the above figure. $\angle B \cong \angle C$

Which is the greater quantity?

(a) $m \widehat{ABC}$

(b) $m \widehat{CAB}$

Answer

To compare $m \widehat{ABC}$ and $m \widehat{CAB}$ , we note that

$m \widehat{ABC} = m \widehat{AB} + m \widehat{BC}$

and

$m \widehat{CAB} = m \widehat{AB} + m \widehat{AC}$

We need to be able to compare $m \widehat{BC}$ and $m \widehat{AC}$ . If we know which of the intercepting angles is the greater, then we know which of the arcs is greater. The intercepting angles are $\angle A , \angle B$ , respectively. However, we are not given this relationship.

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Question

A giant clock has a minute hand that is six feet long. The time is now 3:50 PM. How far has the tip of the minute hand moved, in inches, between noon and now?

Answer

Every hour, the tip of the minute hand travels the circumference of a circle with radius six feet, which is

$C = 2 \pi r= 2 \pi \cdot 6 = 12 \pi$ feet.

Since it is now 3:50 PM, the minute hand made three complete revolutions since noon, plus $\frac{50}{60 }$ = $\frac{5}{6}$ of a fourth, so its tip has traveled this circumference $3$\frac{5}{6}$$ times.

This is

$3$\frac{5}{6}$ \times 12 \pi = $\frac{23}{6}$ \times 12 \pi = 46 \pi$ feet. This is

$46 \pi \times 12 = 552 \pi$ inches.

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Question

A giant clock has a minute hand seven feet long. Which is the greater quantity?

(A) The distance traveled by the tip of the minute hand between 1:30 PM and 2:00 PM

(B) The circumference of a circle seven feet in diameter

Answer

The tip of a minute hand travels a circle whose radius is equal to the length of that minute hand, which, in this question, is seven feet long. The circumference of this circle is $2 \pi$ times the radius, or $2 \pi \times 7 = 14\pi$ feet; over the course of thrity minutes (or one-half of an hour) the tip of the minute hand covers half this distance, or $$\frac{1}{2}$ \times 14 \pi = 7 \pi$ feet.

The circumference of a circle seven feet in diameter is $\pi$ times this diameter, or $7 \pi$ feet.

The quantities are equal.

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Question

A giant clock has a minute hand four and one-half feet in length. Since noon, the tip of the minute hand has traveled $110 \pi$ feet. Which of the following is true of the time right now?

Answer

Every hour, the tip of the minute hand travels the circumference of a circle, which here is

$C = 2 \pi r = 2 \pi \times 4 $\frac{1}{2}$ = 9 \pi$ feet.

The minute hand has traveled $110 \pi$ feet since noon, so it has traveled the circumference of the circle

$\frac{110 \pi}{9 \pi}$ = $\frac{110 }{9 }$ =12 $\frac{2}{9}$ times.

Since $12 \leq 12 $\frac{2}{9}$ \leq 12 $\frac{1}{2}$$ , between 12 and $12 $\frac{1}{2}$$ hours have elapsed since noon, and the time is between 12:00 midnight and 12:30 AM.

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Question

Acute triangle $\bigtriangleup ABC$ is inscribed in a circle. Which is the greater quantity?

(a) $m \overarc {ABC} - m \overarc {ACB}$

(b) $m \overarc {A B} - m \overarc {A C}$

Answer

Examine the figure below, which shows $\bigtriangleup ABC$ inscribed in a circle.

By the Arc Addition Principle,

$m \overarc {ABC} = m \overarc {AB}+ m \overarc {BC}$

and

$m \overarc {ACB} = m \overarc {AC}+ m \overarc {BC}$

Consequently,

$m \overarc {ABC} - m \overarc {ACB}$

$= (m \overarc {AB}+ m \overarc {BC} ) - ( m \overarc {AC}+ m \overarc {BC})$

$= m \overarc {AB}+ m \overarc {BC} - m \overarc {AC}- m \overarc {BC}$

$= m \overarc {A B} - m \overarc {A C}$

The two quantities are equal.

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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 the area of a circle is 1.44 $\pi$ , what is its circumference?

Answer

The answer is $2.4 \pi$ .

Utilizing the formula for area of a circle $\left( \pi $r^{2}\right)$ , you would plug in the answer for area as

$1.44\pi$ = $\pi $r^{2}$$

Divide both sides by $\pi$ .

Then square root both sides to get

Then plug in 1.2 for in the equation for circumference for a circle, $2\pi r$ . Thus

$C = 2.4\pi$

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Question

Find the perimeter around the following semicircle.

Answer

The answer is $9\pi + 18cm$ .

First, you would need to find the radius of the semi-circle. 18 divided by 2 results in 9 cm for the radius. Then you would take the formula for finding circumference $C=2\pi r$ and plug in to get

$C=18\pi$ .

Then you would divide that result by 2 to get $9\pi$ since it is a semicircle. Lastly you would add 18 cm to $9\pi$ because the perimeter is the sum of the semicircle and the diameter. Remember that they are not like terms.

If you chose $9\pi$ , you forgot to include the diameter.

If you chose $27\pi$ , you added and , but they are not like terms.

If you chose $18\pi + 18$ , remember that you only need half of the circumference.

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Question

A sector of a circle with radius of feet has an area of $\small 36\pi$ square feet. Find the length of the arc of the sector.

Answer

We begin with the formula for the area of a sector

$\small S=\frac{m}{360}$\pi $r^2$$

where $\small m$ is the measure of the central angle in degrees and $\small r$ is the radius.

Substituting what we know gives

$\small 36\pi=\frac{m}{360}$\pi $(12^2$)$

$\small 36\pi=\frac{4\pi m}{10}$$

$\small 360\pi=4\pi m$

$\small 90=m$

Therefore our central angle is $\small 90^\circ$

We then turn to our formula for arc length.

$\small s=\frac{m}{360}$2\pi r$

Substituting gives

$\small s=\frac{90}{360}$(2\pi)(12)=\frac{1}{4}$(24\pi)=6\pi$

Therefore, our arc length is $\small 6\pi,ft.$

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Question

The radius of a circle is . Find the length of an arc if it has a measure of degrees.

Answer

Recall that the length of an arc is merely a part of the circle's circumference.

We can then write the following equation to find the length of an arc:

$\text{Arc Length}$=\frac{\text{Arc angle measure}$}{360}\times2\pi\times $\text{radius}$

Plug in the values of the arc angle measure and the radius to find the length of the arc.

$$\text{Arc Length}$=\frac{120}{360}$\times 2\pi\times2=\frac{4}{3}$\pi$

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Question

The radius of a circle is . Find the length of an arc if it has a measurement of degrees.

Answer

Recall that the length of an arc is merely a part of the circle's circumference.

We can then write the following equation to find the length of an arc:

$\text{Arc Length}$=\frac{\text{Arc angle measure}$}{360}\times2\pi\times $\text{radius}$

Plug in the values of the arc angle measure and the radius to find the length of the arc.

$$\text{Arc Length}$=\frac{240}{360}$\times 2\pi\times 3=4\pi$

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Question

The radius of a circle is . Find the length of an arc that has a measurement of degrees.

Answer

Recall that the length of an arc is merely a part of the circle's circumference.

We can then write the following equation to find the length of an arc:

$\text{Arc Length}$=\frac{\text{Arc angle measure}$}{360}\times2\pi\times $\text{radius}$

Plug in the values of the arc angle measure and the radius to find the length of the arc.

$$\text{Arc Length}$=\frac{45}{360}$\times 2\pi\times 4=\pi$

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Question

The radius of a circle is . Find the length of an arc if it has a measure of degrees.

Answer

Recall that the length of an arc is merely a part of the circle's circumference.

We can then write the following equation to find the length of an arc:

$\text{Arc Length}$=\frac{\text{Arc angle measure}$}{360}\times2\pi\times $\text{radius}$

Plug in the values of the arc angle measure and the radius to find the length of the arc.

$$\text{Arc Length}$=\frac{150}{360}$\times 2\pi\times 5=\frac{25}{6}$\pi$

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Question

The radius of a circle is . Find the length of an arc that has a measure of degrees.

Answer

Recall that the length of an arc is merely a part of the circle's circumference.

We can then write the following equation to find the length of an arc:

$\text{Arc Length}$=\frac{\text{Arc angle measure}$}{360}\times2\pi\times $\text{radius}$

Plug in the values of the arc angle measure and the radius to find the length of the arc.

$$\text{Arc Length}$=\frac{95}{360}$\times 2\pi\times 6=\frac{19}{6}$\pi$

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Question

The radius of a circle is . Find the length of an arc if it has a measure of degrees.

Answer

Recall that the length of an arc is merely a part of the circle's circumference.

We can then write the following equation to find the length of an arc:

$\text{Arc Length}$=\frac{\text{Arc angle measure}$}{360}\times2\pi\times $\text{radius}$

Plug in the values of the arc angle measure and the radius to find the length of the arc.

$$\text{Arc Length}$=\frac{30}{360}$\times 2\pi\times 6=\pi$

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Question

The radius of a circle is . Find the length of an arc if it has a measure of degrees.

Answer

Recall that the length of an arc is merely a part of the circle's circumference.

We can then write the following equation to find the length of an arc:

$\text{Arc Length}$=\frac{\text{Arc angle measure}$}{360}\times2\pi\times $\text{radius}$

Plug in the values of the arc angle measure and the radius to find the length of the arc.

$$\text{Arc Length}$=\frac{60}{360}$\times 2\pi\times 8=\frac{2}{3}$\pi$

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Question

The radius of a circle is . Find the length of an arc if it has a measure of degrees.

Answer

Recall that the length of an arc is merely a part of the circle's circumference.

We can then write the following equation to find the length of an arc:

$\text{Arc Length}$=\frac{\text{Arc angle measure}$}{360}\times2\pi\times $\text{radius}$

Plug in the values of the arc angle measure and the radius to find the length of the arc.

$$\text{Arc Length}$=\frac{200}{360}$\times 2\pi\times 9=10\pi$

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Question

The radius of a circle is . Find the length of an arc if it has a measure of degrees.

Answer

Recall that the length of an arc is merely a part of the circle's circumference.

We can then write the following equation to find the length of an arc:

$\text{Arc Length}$=\frac{\text{Arc angle measure}$}{360}\times2\pi\times $\text{radius}$

Plug in the values of the arc angle measure and the radius to find the length of the arc.

$$\text{Arc Length}$=\frac{40}{360}$\times 2\pi\times 10=\frac{20}{9}$\pi$

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Question

The radius of a circle is . Find the length of an arc if it has a measure of degrees.

Answer

Recall that the length of an arc is merely a part of the circle's circumference.

We can then write the following equation to find the length of an arc:

$\text{Arc Length}$=\frac{\text{Arc angle measure}$}{360}\times2\pi\times $\text{radius}$

Plug in the values of the arc angle measure and the radius to find the length of the arc.

$$\text{Arc Length}$=\frac{130}{360}$\times 2\pi\times 12=\frac{26}{3}$\pi$

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