# SAT Math : Sectors

## Example Questions

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### Example Question #1 : Sectors

Two pizzas are made to the same dimensions. The only difference is that Pizza 1 is cut into pieces at 30° angles and Pizza 2 is cut at 45° angles. They are sold by the piece, the first for $1.95 per slice and the second for$2.25 per slice. What is the difference in total revenue between Pizza 2 and Pizza 1?

–$2.70$5.40

$2.70 –$5.40

$0 Correct answer: –$5.40

Explanation:

First, let's calculate how many slices there are per pizza. This is done by dividing 360° by the respective slice degrees:

Pizza 1: 360/30 = 12 slices

Pizza 2: 360/45 = 8 slices

Now, the total amount made per pizza is calculated by multiplying the number of slices by the respective cost per slice:

Pizza 1: 12 * 1.95 = $23.40 Pizza 2: 8 * 2.25 =$18.00

The difference between Pizza 2 and Pizza 1 is thus represented by: 18 – 23.40 = –$5.40 ### Example Question #1 : How To Find The Area Of A Sector A circular, 8-slice pizza is placed in a square box that has dimensions four inches larger than the diameter of the pizza. If the box covers a surface area of 256 in2, what is the surface area of one piece of pizza? Possible Answers: 18π in2 4.5π in2 9π in2 144π in2 36π in2 Correct answer: 4.5π in2 Explanation: The first thing to do is calculate the dimensions of the pizza box. Based on our data, we know 256 = s2. Solving for s (by taking the square root of both sides), we get 16 = s (or s = 16). Now, we know that the diameter of the pizza is four inches less than 16 inches. That is, it is 12 inches. Be careful! The area of the circle is given in terms of radius, which is half the diameter, or 6 inches. Therefore, the area of the pizza is π * 62 = 36π in2. If the pizza is 8-slices, one slice is equal to 1/8 of the total pizza or (36π)/8 = 4.5π in2. ### Example Question #1 : How To Find The Area Of A Sector If B is a circle with line AC = 12 and line BC = 16, then what is the area formed by DBE? Possible Answers: Correct answer: Explanation: Line AB is a radius of Circle B, which can be found using the Pythagorean Theorem: Since AB is a radius of B, we can find the area of circle B via: Angle DBE is a right angle, and therefore of the circle so it follows: ### 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: Explanation: Area of Circle = πr2 = π42 = 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 #1 : How To Find The Area Of A Sector 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?

Explanation:

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 52π = 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 #2 : How To Find The Area Of A Sector

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?

Explanation:

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 #1 : How To Find The Area Of A Sector

A central angle of a circle measures 60 degrees. If its corresponding arc measures 3 units, what is the area of the circle?

Explanation:

If the central angle measures 60 degrees, divide the 360 total degrees in the circle by 60.

Multiply this by the measure of the corresponding arc to find the total circumference of the circle.

Use the circumference to find the radius, then use the radius to find the area.

### 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?

36 + 20π

36 + 10π

18 + 36π

36 + 36π

18 + 10π

36 + 10π

Explanation:

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 2πr, according to the formula for circumference.

length of arc AB = (5/18)(2πr) = (5/18)(2π(18)) = 10π.

Thus, the length of arc AB is 10π.

The total length of the perimeter is thus 36 + 10π.

The answer is 36 + 10π.

### 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?

Explanation:

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 ?

Explanation:

The formula for arclength is .

You know that  so  and  must both equal .

Since

,

the sum the lengths of arcs  and  must equal .

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