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SAT Math : How to find the length of an arc

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Example Question #1 : Circles

Figure not drawn to scale.

In the figure above, circle C has a radius of 18, and the measure of angle ACB is equal to 100°. What is the perimeter of the red shaded region?

Possible Answers:

36 + 36π

18 + 10π

36 + 20π

36 + 10π

18 + 36π

Correct answer:

36 + 10π

Explanation:

The perimeter of any region is the total distance around its boundaries. The perimeter of the shaded region consists of the two straight line segments, AC and BC, as well as the arc AB. In order to find the perimeter of the whole region, we must add the lengths of AC, BC, and the arc AB.

The lengths of AC and BC are both going to be equal to the length of the radius, which is 18. Thus, the perimeter of AC and BC together is 36.

Lastly, we must find the length of arc AB and add it to 36 to get the whole perimeter of the region.

Angle ACB is a central angle, and it intercepts arc AB. The length of AB is going to equal a certain portion of the circumference. This portion will be equal to the ratio of the measure of angle ACB to the measure of the total degrees in the circle. There are 360 degrees in any circle. The ratio of the angle ACB to 360 degrees will be 100/360 = 5/18. Thus, the length of the arc AB will be 5/18 of the circumference of the circle, which equals 2πr, according to the formula for circumference.

length of arc AB = (5/18)(2πr) = (5/18)(2π(18)) = 10π.

Thus, the length of arc AB is 10π.

The total length of the perimeter is thus 36 + 10π.

The answer is 36 + 10π.

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

Circle

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

What is the length of arc A?

Possible Answers:

$\frac{\pi r}{4}$

$\frac{\pi r}{2}$

$\frac{\pi r}{16}$

$\pi r$

$\frac{\pi r}{8}$

Correct answer:

$\frac{\pi r}{4}$

Explanation:

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

In the figure above, $\overline{AC}$ and $\overline{BD}$ are diameters of the cirlce, which has a radius of . What is the sum of the lengths of arcs $\widehat{AB}$ and $\widehat{CD}$ ?

Possible Answers:

$2\pi$

$4\pi$

$2\pi ^{2}$

$24\pi$

$12\pi$

Correct answer:

$4\pi$

Explanation:

The formula for arclength is $\frac{n}{360}* 2\pi r$ .

You know that $\angle AED = 150$ so $\angle BEA$ and $\angle CED$ must both equal .

Since

$\frac{30}{360} * 2\pi (12) = 2\pi$ ,

the sum the lengths of arcs $\widehat{AB}$ and $\widehat{CD}$ must equal $4\pi$ .

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

Secant

Figure NOT drawn to scale

Refer to the above figure. Evaluate .

Possible Answers:

t = 56

t = 28

t = 14

t = 84

t = 21

Correct answer:

t = 28

Explanation:

$\overline{NB}$ and $\overline{ND }$ , being secant segments of a circle, intercept two arcs such that the measure of the angle that the segments form is equal to one-half the difference of the measures of the intercepted arcs - that is,

$\frac{1}{2} (m \overarc{BD} - m \overarc {AC }) = m \angle BND$

Setting $m \overarc{BD} = 3t , m \overarc {AC } = t , m \angle BND = 28^{\circ }$ and solving for :

$\frac{1}{2} (3t - t ) = 28$

$\frac{1}{2} (2t ) = 28$

t= 28

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SAT Math : How to find the length of an arc

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #1 : Circles

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

All SAT Math Resources

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