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Example Questions
Example Question #251 : Problem Solving Questions
A teacher buys a supersized pizza for his after-school club. The super-pizza has a diameter of 18 inches. If the teacher is able to perfectly cut from the center a 36 degree sector for himself, what is the area of his slice of pizza, rounded to the nearest square inch?
28
24
25
26
27
25
First we calculate the area of the pizza. The area of a circle is defined as . Since our diameter is 18 inches, our radius is 18/2 = 9 inches. So the total area of the pizza is square inches.
Since the sector of the pie he cut for himself is 36 degrees, we can set up a ratio to find how much of the pizza he cut for himself. Let x be the area of the pizza he cut for himself. Then we know,
Solving for x, we get x=25.45 square inches, which rounds down to 25.
Example Question #12 : Geometry
In the figure shown below, line segment passes through the center of the circle and has a length of . Points , , and are on the circle. Sector covers of the total area of the circle. Answer the following questions regarding this shape.
Find the area of sector .
To find the area of a sector, we need to know the total area as well as the fractional amount of the sector at which we are looking.
In this case, we find the total area by using the following equation:
Because line segment is our diameter, our radius is . Thus, our total area is:
We need to go one step further to find the area of sector . Simply multiply the total area by the fractional amount that sector covers. We are told it is of the circle's area, so do the following:
Thus, our answer is .
Example Question #1 : Calculating The Area Of A Sector
Consider the Circle :
(Figure not drawn to scale.)
If angle is , what is the area of sector in square meters?
To find the area of a sector, simply multiply the total area of the circle by the fraction of the part you are looking at.
In this case, our area will come from the following:
To find the fractional part of the circle we care about, take the number of degrees in over the total number of degrees in a circle ():
So, we find our answer by multiplying these two parts together:
Example Question #2 : Calculating The Area Of A Sector
If a circle has an area of , what is the area of a sector with an angle of ?
The area of a sector with a certain angle will be whatever fraction of the total circle's area the angle of the sector is of . This means we divide by , and then multiply that fraction by the total area of the circle to give us the area of the sector:
Example Question #2 : Calculating The Area Of A Sector
The circle in the above diagram has center . Give the area of the shaded sector.
Statement 1: Arc has length .
Statement 2: Arc has length .
EITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
Assume Statement 1 alone. Since the circumference of the circle is not given, it cannot be determined what part of the circle is, and therefore, the central angle of the sector cannot be determined. Also, no information about the circle can be determined. A similar argument can be given for Statement 2 being insufficient.
Now assume both statements are true. Then the length of semicircle is equal to . The circumference is twice this, or . The radius can be calculated as , and the area, . Also, is of the circle, and the area of the sector can now be calculated as .
Example Question #252 : Gmat Quantitative Reasoning
Note: Figure NOT drawn to scale
Refer to the above diagram.
What is ?
The degree measure of is half the degree measure of the arc it intercepts, which is . We can use the measures of the two given major arcs to find , then take half of this:
Example Question #253 : Gmat Quantitative Reasoning
A giant clock has a minute hand that is eight feet long. The time is now 2:40 PM. How far has the tip of the minute hand moved, in inches, between noon and now?
Between noon and 2:40 PM, two hours and forty minutes have elapsed, or, equivalently, two and two-thirds hours. This means that the minute hand has made revolutions.
In one revolution, the tip of an eight-foot minute hand moves feet, or inches.
After revolutions, the tip of the minute hand has moved inches.
Example Question #3 : Calculating The Length Of An Arc
In the figure shown below, line segment passes through the center of the circle and has a length of . Points , , and are on the circle. Sector covers of the total area of the circle. Answer the following questions regarding this shape.
What is the length of the arc formed by angle ?
To find arc length, we need to find the total circumference of the circle and then the fraction of the circle we are interested in. Our circumference of a circle formula is:
Where is our radius and is our diameter.
In this problem, our diameter is the length of , which is , so our total circumference is:
Now, to find the fraction of the circle we are interested in, we need to realize that angle is degrees. We know this because it is made by straight line . Armed with this knowledge, we can safely calculate the length of our arc using the following formula:
Example Question #4 : Calculating The Length Of An Arc
Consider the Circle :
(Figure not drawn to scale.)
Suppose is . What is the measure of arc in meters?
To find arc length, multiply the total circumference of a circle by the the fraction of the total circle that defines the arc with the length for which you are solving.
In this case, to find the total circumference:
To find the fraction with which we are concerned, make a fraction with the number of degrees in in the numerator and the total degrees in a circle in the denominator:
Multiply together and simplify:
Example Question #1 : Calculating The Length Of An Arc
What is the arc length for a sector with a central angle of if the radius of the circle is ?
Using the formula for arc length, we can plug in the given angle and radius to calculate the length of the arc that subtends the central angle of the sector. The angle, however, must be in radians, so we make sure to convert degrees accordingly by multiplying the given angle by :