ISEE Upper Level Math : Circles

Study concepts, example questions & explanations for ISEE Upper Level Math

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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?

Possible Answers:

Correct answer:

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!

So our answer is 

Example Question #1 : How To Find The Length Of An Arc

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

Possible Answers:

Correct answer:

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 #1 : Sectors

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

Possible Answers:

It is impossible to tell from the information given

Correct answer:

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 #1 : Sectors

Inscribed

In the above figure, express  in terms of .

Possible Answers:

Correct answer:

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

Intercepted

In the above diagram, radius .

Give the length of .

Possible Answers:

Correct answer:

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?

Possible Answers:

Correct answer:

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

 

So, our answer is 4.54in

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?

Possible Answers:

Correct answer:

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

 Inscribed angle

Figure NOT drawn to scale

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

Possible Answers:

Correct answer:

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

Inscribed angle

Note: Figure NOT drawn to scale

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

Possible Answers:

Correct answer:

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

Intercepted

In the above diagram, radius .

Calculate the length of .

Possible Answers:

Correct answer:

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 .

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