# ISEE Upper Level Math : Circles

## Example Questions

### 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 #81 : Circles

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 #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 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

### Example Question #81 : Circles

A giant clock has a minute hand four feet long. Since noon, the tip of the minute hand has traveled  feet. What time is it now?

Explanation:

The circumference of the path traveled by the tip of the minute hand over the course of one hour is:

feet.

Since the tip of the minute hand has traveled  feet since noon, the minute hand has made

revolutions. Therefore,  hours have elapsed since noon, making the time 1:15 PM.

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

Figure NOT drawn to scale

Refer to the above diagram. is a semicircle. Evaluate  given .

Explanation:

An inscribed angle of a circle that intercepts a semicircle is a right angle; therefore, , which intercepts the semicircle , is such an angle. Consequently,  is a right triangle, and  and  are complementary angles. Therefore,

Inscribed  intercepts an arc with twice its angle measure; this arc is , so

.

The major arc corresponding to this minor arc, , has measure

### Example Question #3 : How To Find The Angle Of A Sector

Note: Figure NOT drawn to scale

Refer to the above diagram. is a semicircle. Evaluate .

Explanation:

An inscribed angle of a circle that intercepts a semicircle is a right angle; therefore, , which intercepts the semicircle , is such an angle. Consequently,

Inscribed  intercepts an arc with twice its angle measure; this arc is , so

.

### Example Question #4 : How To Find The Angle Of A Sector

In the above diagram, radius .

Calculate the length of .

Explanation:

Inscribed , which measures , intercepts an arc with twice its measure. That arc is , which consequently has measure

.

This makes  an arc which comprises

of the circle.

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

.

The length of  is  of this, or .