### All High School Math Resources

## Example Questions

### Example Question #11 : Circles

Find the area of the shaded region:

**Possible Answers:**

**Correct answer:**

To find the area of the shaded region, you must subtract the area of the triangle from the area of the sector.

The formula for the shaded area is:

,

where is the radius of the circle, is the fraction of the sector, is the base of the triangle, and is the height of the triangle.

In order to the find the base and height of the triangle, use the formula for a triangle:

, where is the side opposite the .

Plugging in our final values, we get:

### Example Question #12 : Circles

Find the area of the following sector:

**Possible Answers:**

**Correct answer:**

The formula for the area of a sector is

,

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

Plugging in our values, we get:

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

The radius of the circle above is and . What is the area of the shaded section of the circle?

**Possible Answers:**

**Correct answer:**

Area of Circle = πr^{2} = π4^{2 }= 16π

Total degrees in a circle = 360

Therefore 45 degree slice = 45/360 fraction of circle = 1/8

Shaded Area = 1/8 * Total Area = 1/8 * 16π = 2π

### Example Question #14 : Circles

Find the area of the shaded segment of the circle. The right angle rests at the center of the circle.

**Possible Answers:**

**Correct answer:**

We know that the right angle rests at the center of the circle; thus, the sides of the triangle represent the radius of the circle.

Because the sector of the circle is defined by a right triangle, the region corresponds to one-fourth of the circle.

First, find the total area of the circle and divide it by four to find the area of the depicted sector.

Next, calculate the area of the triangle.

Finally, subtract the area of the triangle from the area of the sector.

### Example Question #15 : Circles

is a square.

The arc from to is a semicircle with a center at the midpoint of .

All units are in feet.

The diagram shows a plot of land.

The cost of summer upkeep is $2.50 per square foot.

In dollars, what is the total upkeep cost for the summer?

**Possible Answers:**

**Correct answer:**

To solve this, we must begin by finding the area of the diagram, which is the area of the square less the area of the semicircle.

The area of the square is straightforward:

30 * 30 = 900 square feet

Because each side is 30 feet long, AB + BC + CD = 30.

We can substitute BC for AB and CD since all three lengths are the same:

BC + BC + BC = 30

3BC = 30

BC = 10

Therefore the diameter of the semicircle is 10 feet, so the radius is 5 feet.

The area of the semi-circle is half the area of a circle with radius 5. The area of the full circle is 5^{2}π = 25π, so the area of the semi-circle is half of that, or 12.5π.

The total area of the plot is the square less the semicircle: 900 - 12.5π square feet

The cost of upkeep is therefore 2.5 * (900 – 12.5π) = $(2250 – 31.25π).

### Example Question #16 : Circles

In the figure, PQ is the arc of a circle with center O. If the area of the sector is what is the perimeter of sector?

**Possible Answers:**

**Correct answer:**

First, we figure out what fraction of the circle is contained in sector OPQ: , so the total area of the circle is .

Using the formula for the area of a circle, , we can see that .

We can use this to solve for the circumference of the circle, , or .

Now, OP and OQ are both equal to *r*, and PQ is equal to of the circumference of the circle, or .

To get the perimeter, we add OP + OQ + PQ, which give us .

### Example Question #17 : Circles

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 #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*

18 + 10*π*

18 + 36*π*

36 + 20*π*

36 + 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 #18 : Circles

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 #19 : Circles

**Possible Answers:**

**Correct answer:**

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