Unlocking the Secrets of Sector Area in Circles

Find the area of the shaded sector formed by $∠ A B C .$ Answer in exact form.

sector ABC with a central angle of 60 degrees and a radius of 10

A central angle of 250 degrees and a radius of 10

green sector of a circle with a central angle of 130 degrees and a radius of 12

A sector of a circle is the region that is bounded by an arc of the circle and the circle's radii that connects to the arc's endpoints.

To calculate the area, we multiply the area of the circle with the ratio of the measure of the central angle to 360^(∘). A=θ/360^(∘)* π r^2 Let's have a look at the given sector.

The sector has a central angle with a measure of 60^(∘) and its radius is 10. With this information, we can determine the area of the sector.

As in Part A, we will use the formula for finding the area of a sector.

As in previous parts, we will use the formula for calculating the sector's area.

What is the radius of the circle if the given area refers to the shaded sector of the circle? Round to the nearest centimeter.

sector of a circle with an area of 453.78 square cm and a central angle of 130 degrees

Diagram showing a sector of a circle with an area of 339.29 square cm and a central angle of 270 degrees

To calculate the area of a sector of a circle, we multiply the area of the circle with the ratio of the central angle to 360^(∘). A=θ/360^(∘)* π r^2 In the given sector, the central angle is 130^(∘) and the corresponding area of the sector is 453.78 cm^2. By substituting this into the formula, we can solve for the radius of the circle.

The radius is about 20 centimeters.

Notice that the right angle is an adjacent angle to the central angle we want to find. To determine the measure of the central angle for our sector, we can subtract 90^(∘) from 360^(∘). 360^(∘)-90^(∘)= 270^(∘) The central angle that relates to the area of the sector is 270 ^(∘).

Now we can use the formula for the area of a sector of a circle and solve for r.

The radius is about 12 centimeters.

The light from a lighthouse covers an area of $700 miles^{2} .$

light emitted from a lighthouse in a 245 degree arc creating a sector of a circle with an area of 700 square miles

How far away from the lighthouse is the light visible? Round to the nearest integer.

To determine the area of a sector of a circle, we must calculate the product of the circle's area and the ratio of the central angle to 360^(∘). A=θ/360^(∘)* π r^2 Examining the diagram, we see that the lighthouse emits light in a sector with a central angle of 245^(∘). We also know that the area of this sector is 700 miles^2. If we substitute this into the given formula, we can solve for the sector's radius.

The sector lit by the lighthouse has a radius of 18 miles. Any ship at this distance or less would be able to see the light.

What is the measure of $A C ?$ Round the answer to the nearest integer.

The area of the sector created by AC can be calculated with the following formula. A=mAC/360^(∘)* π r^2 For the given sector, we know the area and radius. If we substitute these values into the formula, we can solve for the measure of the arc.

As in Part A, we will substitute the area and the radius into the formula for the area of a sector and then solve for the measure of the arc.

If a circle with the radius

r

has a sector with an area

A,

which of the following expressions equal the measure of the sector's central angle?

A . C . E . \frac{3 6 0 ^{\circ} π}{A r ^{2}} 36 0^{\circ} r^{2} (\frac{A}{π}) 36 0^{\circ} A (\frac{1}{π r ^{2}}) B . \frac{3 6 0 ^{\circ} A}{π r ^{2}} D . \frac{A π r ^{2}}{3 6 0 ^{\circ}} F . \frac{A r ^{2}}{3 6 0 ^{\circ} π}

The area of a sector of a circle is the product of the circle's area and the ratio of the measure of the central angle to 360^(∘). A=θ/360^(∘)* π r^2 To determine which of the options are correct, we will solve this equation for θ.

As we can see, B is correct. However, we could also rewrite this as a multiplication. 360^(∘) A/π r^2 = 360^(∘) A(1/π r^2) That means E is also correct.

For each circle, determine the ratio of the shaded sector to the circle. Write the ratio in its simplest form.

four different circles where sectors occupy different ratios of each circle

To determine the ratio that each sector occupies of its circle, we will divide the sector's central angle by 360^(∘). For C and D, we are given the value of the sector's central angle. However, for A and B, we only have their explementary angle. To determine the central angles of the shaded sector of A and B, we will subtract their respective explementary angle measures from 360^(∘). &360^(∘)-60^(∘) =300^(∘) &360^(∘)-320^(∘) =40^(∘) In circles A and B, let's label the central angle that relates to the shaded sector.

Now we can set up ratios and reduce them to their simplest form.

Circle	θ/360^(∘)	Reduce
A	300^(∘)/360^(∘)	5/6
B	40^(∘)/360^(∘)	1/9
C	130^(∘)/360^(∘)	13/36
D	30^(∘)/360^(∘)	1/12

Finally, we will pair each circle with the correct ratio. A&: 5/6 &&B: 1/9 [1em] C&: 13/36 &&D: 1/12

Area of a Sector

Catch-Up and Review

Investigating the Area of a Slice of Pizza

Investigating the Area of Sectors of a Circle

Sector of a Circle

Area of a Sector of a Circle

Proof

Using Sectors of Circles to Solve Problems

Hint

Solution

Calculating the Central Angle of Pac-Man's Mouth

Hint

Solution

Comparing the Radius of Two Circles

Hint

Solution

Using Areas Of Sectors to Solve Problems

Hint

Solution

Solving Real Life Problems Using Sectors of Circles

Hint

Solution

Area of a Sector

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