### All PSAT Math Resources

## Example Questions

### 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 + 10*π*

36 + 10*π*

18 + 36*π*

36 + 36*π*

36 + 20*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 #513 : Geometry

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

### Example Question #1 : Plane Geometry

If the area of a circle is , then what is the length of the arc shown in the diagram?

**Possible Answers:**

**Correct answer:**

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

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

Divide both sides by

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

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.

When you multiply both sides by , you find the solution:

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