### All High School 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 + 36*π*

36 + 20*π*

36 + 36*π*

36 + 10*π*

18 + 10*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 #1 : 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

### Example Question #1 : How To Find The Angle Of A Sector

**Possible Answers:**

**Correct answer:**

We can solve for the angle, , by using the below relationship.

In the figure, intercepted arc is given.

### Example Question #1 : How To Find The Angle Of A Sector

A pie has a diameter of 12". A piece is cut out, having a surface area of 4.5π. What is the angle of the cut?

**Possible Answers:**

12.5°

4.5°

25°

90°

45°

**Correct answer:**

45°

This is simply a matter of percentages. We first have to figure out what percentage of the surface area is represented by 4.5π. To do that, we must calculate the total surface area. If the diameter is 12, the radius is 6. Don't be tricked by this!

A = π * 6 * 6 = 36π

Now, 4.5π is 4.5π/36π percentage or 0.125 (= 12.5%)

To figure out the angle, we must take that percentage of 360°:

0.125 * 360 = 45°

### Example Question #2 : How To Find The Angle Of A Sector

Eric is riding a Ferris wheel. The Ferris wheel has 18 compartments, numbered in order clockwise. If compartment 1 is at 0 degrees and Eric enters compartment 13, what angle is he at?

**Possible Answers:**

280

300

260

240

180

**Correct answer:**

240

12 compartments further means 240 more degrees. 240 is the answer.

360/12 = 240 degrees

### Example Question #1 : How To Find The Angle Of A Sector

**Possible Answers:**

**Correct answer:**

The measure of an arc of an inscribed angle is twice the size of the inscribed angle.

### Example Question #3 : How To Find The Angle Of A Sector

In the figure above that includes Circle O, the measure of angle BAC is equal to 35 degrees, the measure of angle FBD is equal to 40 degrees, and the measure of arc AD is twice the measure of arc AB. Which of the following is the measure of angle CEF? The figure is not necessarily drawn to scale, and the red numbers are used to mark the angles, not represent angle measures.

**Possible Answers:**

**Correct answer:**

The measure of angle CEF is going to be equal to half of the difference between the measures two arcs that it intercepts, namely arcs AD and CD.

Thus, we need to find the measure of arcs AD and CD. Let's look at the information given and determine how it can help us figure out the measures of arcs AD and CD.

Angle BAC is an inscribed angle, which means that its meausre is one-half of the measure of the arc that it incercepts, which is arc BC.

Thus, since angle BAC is 35 degrees, the measure of arc BC must be 70 degrees.

We can use a similar strategy to find the measure of arc CD, which is the arc intercepted by the inscribed angle FBD.

Because angle FBD has a measure of 40 degrees, the measure of arc CD must be 80 degrees.

We have the measures of arcs BC and CD. But we still need the measure of arc AD. We can use the last piece of information given, along with our knowledge about the sum of the arcs of a circle, to determine the measure of arc AD.

We are told that the measure of arc AD is twice the measure of arc AB. We also know that the sum of the measures of arcs AD, AB, CD, and BC must be 360 degrees, because there are 360 degrees in a full circle.

Because AD = 2AB, we can substitute 2AB for AD.

This means the measure of arc AB is 70 degrees, and the measure of arc AD is 2(70) = 140 degrees.

Now, we have all the information we need to find the measure of angle CEF, which is equal to half the difference between the measure of arcs AD and CD.

Certified Tutor

Certified Tutor