### All High School Math Resources

## Example Questions

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

If a quarter of the area of a circle is , then what is a quarter of the circumference of the circle?

**Possible Answers:**

**Correct answer:**

If a quarter of the area of a circle is , then the area of the whole circle is . This means that the radius of the circle is 6. The diameter is 12. Thus, the circumference of the circle is . One fourth of the circumference is .

### Example Question #2 : Sectors

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 + 10*π*

18 + 36*π*

18 + 10*π*

36 + 20*π*

36 + 36*π*

**Correct answer:**

36 + 10*π*

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 2*πr*, according to the formula for circumference.

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

Thus, the length of arc *AB* is 10*π*.

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

The answer is 36 + 10*π*.

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

Find the arc length of a sector that has an angle of 120 degrees and radius of 3.

**Possible Answers:**

**Correct answer:**

The equation for the arc length of a sector is .

Substitute the given radius for and the given angle for to get the following equation:

Simplify:

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

**Possible Answers:**

**Correct answer:**

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

Find the circumference of the following sector:

**Possible Answers:**

**Correct answer:**

The formula for the circumference of a sector is

,

where is the radius of the sector and is the fraction of the sector.

Plugging in our values, we get:

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

In the circle above, the angle A in radians is

What is the length of arc A?

**Possible Answers:**

**Correct answer:**

Circumference of a Circle =

Arc Length

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