### All PSAT 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 + 10*π*

36 + 10*π*

36 + 36*π*

18 + 36*π*

36 + 20*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 #1 : Plane Geometry

If the area of a circle is , then what is the length of the arc shown in the diagram?

**Possible Answers:**

**Correct answer:**

We are given the area of the circle, but we need to find the circumference in order to find the arc length. The equation for the area of a circle is

Because we know that the area is 36, we can use that equation to find the radius of the circle.

Divide both sides by

Take the square root of both sides, and see that the radius is 6.

We can now find the circumference of the circle using the formula

Now that we know the circumference, we can set up a proportion. The length of the 120 degree arc is going to be only a portion of the total circumference of the circle. By putting the degree measure over 360 and setting it equal to x over the circumference, we can find exactly how long the arc is.

When you multiply both sides by , you find the solution:

### 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 in^{2}, what is the surface area of one piece of pizza?

**Possible Answers:**

9π in^{2}

4.5π in^{2}

144π in^{2}

18π in^{2}

36π in^{2}

**Correct answer:**

4.5π in^{2}

The first thing to do is calculate the dimensions of the pizza box. Based on our data, we know 256 = s^{2}. 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 * 6^{2} = 36π in^{2}. If the pizza is 8-slices, one slice is equal to 1/8 of the total pizza or (36π)/8 = 4.5π in^{2}.

### 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:**

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:**

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 #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?

**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 #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?

**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 #1 : Circles

What is the area of the sector above if the radius of circle is ?

**Possible Answers:**

**Correct answer:**

To find the area of a sector, first find the area of the whole circle.

The radius of the circle is 9, so

which can be reduced to .

The area of the sector is only a portion of the total area.To find out exactly how large the area is, set up a proportion where one side equals the angle measure over 360

Multiply both sides by 81 and you will solve for x, which equals

### Example Question #1 : Circles

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?

**Possible Answers:**

$0

$5.40

–$5.40

–$2.70

$2.70

**Correct answer:**

–$5.40

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