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

## Example Questions

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### Example Question #52 : Circles

Which is the greater quantity?

(a) The degree measure of a 10-inch-long arc on a circle with radius 8 inches.

(b) The degree measure of a 12-inch-long arc on a circle with radius 10 inches.

(a) and (b) are equal.

It is impossible to tell from the information given.

(a) is greater.

(b) is greater.

(a) is greater.

Explanation:

(a) A circle with radius 8 inches has crircumference  inches. An arc 10 inches long is  of that circle. , the degree measure of this arc.

(b) A circle with radius 10 inches has crircumference  inches. An arc 12 inches long is  of that circle. , the degree measure of this arc.

(a) is the greater quantity.

### Example Question #53 : Circles

Note: figure NOT drawn to scale

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

(a)

(b)

(a) is greater

(It is impossible to tell from the information given

(b) is greater

(a) and (b) are equal

(a) and (b) are equal

Explanation:

Since

,

the triangle is a right triangle with right angle .

is an inscribed angle on the circle, so the arc it intercepts is a semicricle. Therefore,  is also a semicircle, and it measures .

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

Note: Figure NOT drawn to scale

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

(a)

(b) 90

(b) is greater

(a) is greater

It is impossible to tell from the information given

(a) and (b) are equal

(b) is greater

Explanation:

The measure of an arc intercepted by an inscribed angle of a circle is twice that of the angle. Therefore,

### Example Question #55 : Circles

has twice the radius of . Sector 1 is part of ; Sector 2 is part of ; the two sectors are equal in area.

Which is the greater quantity?

(a) Twice the degree measure of the central angle of Sector 1

(b) The degree measure of the central angle of Sector 2

(b) is greater

It is impossible to tell from the information given

(a) and (b) are equal

(a) is greater

(b) is greater

Explanation:

has twice the radius of , so  has four times the area of . This means that for a sector of  to have the same area as a sector of , the central angle of the latter sector must be four times that of the former sector. This makes (b) greater than (a), which is only twice that of the former sector.

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

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

(a)

(b) 55

(b) is greater

(a) and (b) are equal

(a) is greater

It is impossible to tell from the information given

(a) and (b) are equal

Explanation:

The measure of an inscribed angle of a circle is one-half that of the arc it intercepts. Therefore, .

### Example Question #57 : Circles

The arc-length for the shaded sector is .  What is the value of , rounded to the nearest hundredth?

˚

˚

˚

˚

˚

˚

Explanation:

Remember that the angle for a sector or arc is found as a percentage of the total  degrees of the circle.  The proportion of  to  is the same as  to the total circumference of the circle.

The circumference of a circle is found by:

For our data, this means:

Now we can solve for  using the proportions:

Cross multiply:

Divide both sides by :

Therefore,  is ˚.

### Example Question #58 : Circles

The arc-length for the shaded sector is .  What is the value of , rounded to the nearest hundredth?

˚

˚

˚

˚

˚

˚

Explanation:

Remember that the angle for a sector or arc is found as a percentage of the total  degrees of the circle.  The proportion of  to  is the same as  to the total circumference of the circle.

The circumference of a circle is found by:

For our data, this means:

Now we can solve for  using the proportions:

Cross multiply:

Divide both sides by :

Therefore,  is ˚.

### Example Question #59 : Circles

The area of the shaded sector in circle O is .  What is the angle measure , rounded to the nearest hundredth?

˚

˚

˚

˚

˚

˚

Explanation:

Remember that the angle for a sector or arc is found as a percentage of the total  degrees of the circle.  The proportion of  to  is the same as  to the total area of the circle.

The area of a circle is found by:

For our data, this means:

Now we can solve for  using the proportions:

Cross multiply:

Divide both sides by :

Therefore,  is ˚.

### Example Question #60 : Circles

The area of the shaded sector in circle O is .  What is the angle measure ?

˚

˚

˚

˚

˚

˚

Explanation:

Remember that the angle for a sector or arc is found as a percentage of the total  degrees of the circle.  The proportion of  to  is the same as  to the total area of the circle.

The area of a circle is found by:

For our data, this means:

Now we can solve for  using the proportions:

Cross multiply:

Divide both sides by :

Therefore,  is ˚.

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

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

(a) The area of

(b) The area of the orange semicircle

(b) is the greater quantity

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

(a) is the greater quantity

(a) and (b) are equal

(b) is the greater quantity

Explanation:

has angles of degree measure 30 and 60; 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 smaller leg of a 30-60-90 triangle has length equal to  times that of the longer leg; this is about

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:

The orange semicircle has a greater area than

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