ISEE Upper Level Quantitative : How to find the area of a sector

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

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Example Questions

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

Circle A has twice the radius of Circle B. Which is the greater quantity?

(a) The area of a  sector of Circle A

(b) The area of Circle B

Possible Answers:

(a) and (b) are equal.

(a) is greater.

It is impossible to tell from the information given.

(b) is greater.

Correct answer:

(a) and (b) are equal.

Explanation:

Let  be the radius of Circle B. The radius of Circle A is therefore .

 sector of a circle comprises  of the circle. The  sector of circle A has area , the area of Circle B. The two quantities are equal.

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

Generalsector-12

What is the area, rounded to the nearest hundredth, of the sector shaded in circle O in the diagram above?

Possible Answers:

Correct answer:

Explanation:

To find the area of a sector, you need to find a percentage of the total area of the circle.  You do this by dividing the sector angle by the total number of degrees in a full circle (i.e. ˚).  Thus, for our circle, which has a sector with an angle of ˚, we have a percentage of:

Now, we will multiply this by the total area of the circle.  Recall that we find such an area according to the equation:

For our problem,

Therefore, our equation is:

 

Using your calculator, you can determine that this is approximately .

Example Question #6 : Sectors

Generalsector-12

What is the area, rounded to the nearest hundredth, of the sector shaded in circle O in the diagram above?

Possible Answers:

Correct answer:

Explanation:

To find the area of a sector, you need to find a percentage of the total area of the circle.  You do this by dividing the sector angle by the total number of degrees in a full circle (i.e. ˚).  Thus, for our circle, which has a sector with an angle of ˚, we have a percentage of:

Now, we will multiply this by the total area of the circle.  Recall that we find such an area according to the equation:

For our problem, 

Therefore, our equation is:

 

Using your calculator, you can determine that this is approximately .

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

Icecreamcone 2

Refer to the above figure, Which is the greater quantity?

(a) The area of 

(b) The area of the orange semicircle 

Possible Answers:

It is impossible to determine which is greater from the information given

(a) and (b) are equal

(b) is the greater quantity

(a) is the greater quantity

Correct answer:

(a) is the greater quantity

Explanation:

 has two angles of degree measure 45; the third angle must measure 90 degrees, making  a right triangle.

For the sake of simplicity, let ; the reasoning is independent of the actual length. The legs of a 45-45-90 triangle are congruent, so . The area of a right triangle is half the product of its legs, so 

Also, if , then the orange semicircle has diameter 1 and radius . Its area can be found by substituting  in the formula:

 has a greater area than the orange semicircle.

Example Question #8 : Sectors

Refer to the above figure, Which is the greater quantity?

(a) The area of the orange semicircle 

(b) The area of 

Possible Answers:

(b) is the greater quantity

(a) is the greater quantity

It is impossible to determine which is greater from the information given

(a) and (b) are equal

Correct answer:

(b) is the greater quantity

Explanation:

 has two angles of degree measure 60; its third angle must also have measure 60, making  an equilateral triangle 

For the sake of simplicity, let ; the reasoning is independent of the actual length. The area of equilateral  can be found by substituting  in the formula

Also, if , then the orange semicircle has diameter 1 and radius . Its area can be found by substituting  in the formula:

 

 has a greater area than the orange semicircle.

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

Circle 1

The above circle, which is divided into sectors of equal size, has diameter 20. Give the area of the shaded region.

Possible Answers:

Correct answer:

Explanation:

The radius of a circle is half its diameter; the radius of the circle in the diagram is half of 20, or 10.

To find the area of the circle, set  in the area formula:

The circle is divided into sixteen sectors of equal size, five of which are shaded; the shaded portion is

.

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