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PSAT Math : How to find the length of an arc

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

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:

18 + 36π

36 + 36π

36 + 20π

36 + 10π

18 + 10π

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:

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

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

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

$\pi r$

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

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 #3 : Plane Geometry

Circle_120_degrees

If the area of a circle is $36\pi$ , then what is the length of the $120^\circ$ arc shown in the diagram?

Possible Answers:

$8\pi$

$4\pi$

$6\pi$

$12\pi$

Correct answer:

$4\pi$

Explanation:

We are given the area of the circle, but we need to find the circumference in order to find the arc length. The equation for the area of a circle is $A =\pi r^{2}$

Because we know that the area is 36 $\pi$ , we can use that equation to find the radius of the circle.

$36\pi =\pi r^{2}$

Divide both sides by $\pi$

$36 = r^{2}$

Take the square root of both sides, and see that the radius is 6.

We can now find the circumference of the circle using the formula $C=2\pi r$

$C=2\pi 6$

$C=12\pi$

Now that we know the circumference, we can set up a proportion. The length of the 120 degree arc is going to be only a portion of the total circumference of the circle. By putting the degree measure over 360 and setting it equal to x over the circumference, we can find exactly how long the arc is.

$\frac{120}{360}=\frac{x}{12\pi }$

When you multiply both sides by $12\pi$ , you find the solution:

$x=4\pi$

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PSAT Math : How to find the length of an arc

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

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

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

Example Question #3 : Plane Geometry

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