Sectors - PSAT Math

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_.

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:

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

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).

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.

First, we figure out what fraction of the circle is contained in sector OPQ: $$\frac{30^{circ}$$$}{360^{circ}$}= $\frac{1}{12}$, so the total area of the circle is 12times 3pi=36 .

Using the formula for the area of a circle, ${pi}r^{2}$, we can see that r=6.

We can use this to solve for the circumference of the circle, 2{pi}r, or 12{pi}.

Now, OP and OQ are both equal to r, and PQ is equal to $\frac{1}{12}$ of the circumference of the circle, or {pi}.

To get the perimeter, we add OP + OQ + PQ, which give us 12+{pi}.

Line AB is a radius of Circle B, which can be found using the Pythagorean Theorem:

$AB^2$$=AC^2$$+BC^2rightarrow$ $AB=$\sqrt{AC^2$$+BC^2$$}$=$\sqrt{16^2$$+12^2$}$=$\sqrt{400}$=20

Since AB is a radius of B, we can find the area of circle B via:

Area=pi $R^2$$=pi(20^2$)=400pi

Angle DBE is a right angle, and therefore of the circle so it follows:

Area(DBE)=\frac{400}{4}$pi=100pi

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

Circumference of a Circle =

Arc Length

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_.

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:

Circumference of a Circle =

Arc Length

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_.

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:

Since we know that segment AC is a diameter, this means that the length of the arc ABC must be 180 degrees. This means that the length of the arc AB must be 80 degrees.

Since angle ADB is an inscribed angle, its measure is equal to half of the measure of the angle of the arc that it intercepts. This means that the measure of the angle is half of 80 degrees, or 40 degrees.

The circumference of the circle is .

The length of the arc S is .

A ratio can be established:

Solving for __yields 90o.

Note: This makes sense. Since the arc S was one-fourth the circumference of the circle, the central angle formed by arc S should be one-fourth the total degrees of a circle.

The total area of the circle is the sum of the areas of the white and gray sectors, or

The gray sector takes up

of the circle, so the degree measure of the gray sector is equal to

The measure of angle CEF is going to be equal to half of the difference between the measures two arcs that it intercepts, namely arcs AD and CD.

Thus, we need to find the measure of arcs AD and CD. Let's look at the information given and determine how it can help us figure out the measures of arcs AD and CD.

Angle BAC is an inscribed angle, which means that its meausre is one-half of the measure of the arc that it incercepts, which is arc BC.

Thus, since angle BAC is 35 degrees, the measure of arc BC must be 70 degrees.

We can use a similar strategy to find the measure of arc CD, which is the arc intercepted by the inscribed angle FBD.

Because angle FBD has a measure of 40 degrees, the measure of arc CD must be 80 degrees.

We have the measures of arcs BC and CD. But we still need the measure of arc AD. We can use the last piece of information given, along with our knowledge about the sum of the arcs of a circle, to determine the measure of arc AD.

We are told that the measure of arc AD is twice the measure of arc AB. We also know that the sum of the measures of arcs AD, AB, CD, and BC must be 360 degrees, because there are 360 degrees in a full circle.

Because AD = 2AB, we can substitute 2AB for AD.

This means the measure of arc AB is 70 degrees, and the measure of arc AD is 2(70) = 140 degrees.

Now, we have all the information we need to find the measure of angle CEF, which is equal to half the difference between the measure of arcs AD and CD.

This is simply a matter of percentages. We first have to figure out what percentage of the surface area is represented by 4.5π. To do that, we must calculate the total surface area. If the diameter is 12, the radius is 6. Don't be tricked by this!

A = π * 6 * 6 = 36π

Now, 4.5π is 4.5π/36π percentage or 0.125 (= 12.5%)

To figure out the angle, we must take that percentage of 360°:

0.125 * 360 = 45°