ACT Math : How to find the length of an arc

Study concepts, example questions & explanations for ACT Math

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Example Questions

Example Question #513 : Geometry

Circle

In the circle above, the angle A in radians is 

What is the length of arc A?

Possible Answers:

Correct answer:

Explanation:

Circumference of a Circle = 

Arc Length

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

_arc2

The area of circle O is . The area of the shaded sector is . This figure is not drawn to scale.

What is the length of the arc ?

Possible Answers:

Correct answer:

Explanation:

If the area of the circle is , the radius can be found using the formula for the area of a circle:

For our data, this is:

Therefore, 

Now, the circumference of the circle is defined as:

For our data, this is:

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

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

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

_arc1

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?

Possible Answers:

Correct answer:

Explanation:

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

For our data, this is:

Now the percentage for our arc is based on the angle  and the total degrees in a circle, namely, .

So, the length of the arc is:

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

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? 

Possible Answers:

Correct answer:

Explanation:

A circle has a total of  degrees. If our angle is located at the center and is  degrees, we can do  to see that our angle makes up  of the complete circle.

Therefore, our arc is going to be  of our total circumference. 

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

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 .

Possible Answers:

Correct answer:

Explanation:

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

The central angle given is 120 thus:

Example Question #4 : How To Find The Length Of An Arc

A water wheel turns a  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:

Correct answer:

Explanation:

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

 

If the wheel turns  each minute, then it turns  of the circumference each minute.

Thus, the wheel turns  each minute.

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