# ISEE Upper Level Math : Sectors

## Example Questions

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### Example Question #1 : How To Find The Angle For A Percentage Of A Circle

Sector TYP occupies 43% of a circle. Find the degree measure of angle TYP.

Explanation:

Sector TYP occupies 43% of a circle. Find the degree measure of angle TYP.

Use the following formula and solve for x:

Begin by dividing over the 100

Then multiply by 360

### Example Question #2 : How To Find The Angle For A Percentage Of A Circle

If sector AJL covers 45% of circle J, what is the measure of sector AJL's central angle?

Explanation:

If sector AJL covers 45% of circle J, what is the measure of sector AJL's central angle?

To find an angle measure from a percentage, simply convert the percentage to a decimal and then multiply it by 360 degrees.

So, our answer is 162 degrees.

### Example Question #1 : How To Find The Area Of A Sector

Give the area of the white region of the above circle if  has length

Explanation:

If we let  be the circumference of the circle, then the length of  is  of the circumference, so

The radius is the circumference divided by :

Use the formula to find the area of the entire circle:

The area of the white region is  of that of the circle, or

### Example Question #2 : How To Find The Area Of A Sector

The circumference of the above circle is . Give the area of the shaded region.

Explanation:

The radius of a circle is found by dividing the circumference  by :

The area of the entire circle can be found by substituting for  in the formula:

.

The area of the shaded  sector is  of the total area:

### Example Question #1 : Sectors

While visiting a history museum, you see a radar display which consists of a circular screen with a highlighted wedge with an angle of . If the screen has a radius of 4 inches, what is the area of the highlighted wedge?

Explanation:

While visiting a history museum, you see a radar display which consists of a circular screen with a highlighted wedge with an angle of . If the screen has a radius of 4 inches, what is the area of the highlighted wedge?

To begin, let's recall our formula for area of a sector.

Now, we have theta and r, so we just need to plug them in and simplify!

### Example Question #211 : Plane Geometry

A giant clock has a minute hand six feet long. How far, in inches, did the tip move between noon and 1:20 PM?

Explanation:

The distance that the tip of the minute hand moves during one hour is the circumference of a circle with radius 6 feet. This circumference is  feet. One hour and twenty minutes is  hours, so the tip of the hand moved  feet, or  inches.

### Example Question #2 : How To Find The Length Of An Arc

A giant clock has a minute hand three feet long. How far, in inches, did the tip move between noon and 12:20 PM?

It is impossible to tell from the information given

Explanation:

The distance that the tip of the minute hand moves during one hour is the circumference of a circle with radius  feet. This circumference is  feet.  minutes is one-third of an hour, so the tip of the minute hand moves  feet, or  inches.

### Example Question #3 : How To Find The Length Of An Arc

In the above figure, express  in terms of .

Explanation:

The measure of an arc -  - intercepted by an inscribed angle -  - is twice the measure of that angle, so

### Example Question #4 : How To Find The Length Of An Arc

In the above diagram, radius .

Give the length of .

Explanation:

The circumference of a circle is  multiplied by its radius , so

.

, being an inscribed angle of the circle, intercepts an arc  with twice its measure:

The length of  is the circumference multiplied by :

.

### Example Question #5 : How To Find The Length Of An Arc

While visiting a history museum, you see a radar display which consists of a circular screen with a highlighted wedge with an angle of . If the screen has a radius of 4 inches, what is the length of the arc of the highlighted wedge?

Explanation:

While visiting a history museum, you see a radar display which consists of a circular screen with a highlighted wedge with an angle of . If the screen has a radius of 4 inches, what is the length of the arc of the highlighted wedge?

To begin, let's recall our formula for length of an arc.

Now, just plug in and simplify