### All SAT 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:**

36 + 36*π*

36 + 20*π*

18 + 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 #1 : Sectors

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 #3 : How To Find The Length Of An Arc

In the figure above, and are diameters of the cirlce, which has a radius of . What is the sum of the lengths of arcs and ?

**Possible Answers:**

**Correct answer:**

The formula for arclength is .

You know that so and must both equal .

Since

,

the sum the lengths of arcs and must equal .

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

Figure NOT drawn to scale

Refer to the above figure. Evaluate .

**Possible Answers:**

**Correct answer:**

and , being secant segments of a circle, intercept two arcs such that the measure of the angle that the segments form is equal to one-half the difference of the measures of the intercepted arcs - that is,

Setting and solving for :