# ISEE Upper Level Quantitative : How to find the length of a radius

## Example Questions

### Example Question #1 : How To Find The Length Of A Radius

The area of Circle B is four times that of Circle A. The area of Circle C is four times that of Circle B. Which is the greater quantity?

(a) Twice the radius of Circle B

(b) The sum of the radius of Circle A and the radius of Circle C

It cannot be determined from the information given.

(b) is greater.

(a) is greater.

(a) and (b) are equal.

(b) is greater.

Explanation:

Let  be the radius of Circle A. Then its area is .

The area of Circle B is , so the radius of Circle B is twice that of Circle A; by a similar argument, the radius of Circle C is twice that of Circle B, or .

(a) Twice the radius of circle B is .

(b) The sum of the radii of Circles A and B is .

This makes (b) greater.

### Example Question #2 : How To Find The Length Of A Radius

The time is now 1:45 PM. Since noon, the tip of the minute hand of a large clock has moved  feet. How long is the minute hand of the clock?

Explanation:

Every hour, the tip of the minute hand travels the circumference of a circle. Between noon and 1:45 PM, one and three-fourths hours pass, so the tip travels  or  times this circumference. The length of the minute hand is the radius of this circle , and the circumference of the circle is , so the distance the tip travels is  this, or

Set this equal to  feet:

feet.

This is equivalent to 1 foot 4 inches.

### Example Question #1 : Radius

The tip of the minute hand of a giant clock has traveled  feet since noon. It is now 2:30 PM. Which is the greater quantity?

(A) The length of the minute hand

(B) Three yards

(A) and (B) are equal

(A) is greater

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

(B) is greater

(B) is greater

Explanation:

Betwen noon and 2:30 PM, the minute hand has made two and one-half revolutions; that is, the tip of minute hand has traveled the circumference of its circle two and one-half times. Therefore,

feet.

The radius of this circle  is the length of the minute hand. We can use the circumference formula to find this:

The minute hand is eight feet long, which is less than three yards (nine feet), so (B0 is greater.

### Example Question #4 : How To Find The Length Of A Radius

Compare the two quantities:

Quantity A: The radius of a circle with area of

Quantity B: The radius of a circle with circumference of

The two quantities are equal.

The quantity in Column A is greater.

The quantity in Column B is greater.

The relationship cannot be determined from the information given.

The quantity in Column B is greater.

Explanation:

Recall for this question that the formulae for the area and circumference of a circle are, respectively:

For our two quantities, we have:

Quantity A

Therefore,

Taking the square root of both sides, we get:

Quantity B

Therefore,

Therefore, quantity B is greater.

### Example Question #5 : How To Find The Length Of A Radius

Compare the two quantities:

Quantity A: The radius of a circle with area of

Quantity B: The radius of a circle with circumference of

The relationship cannot be determined from the information given.

The quantity in Column B is greater.

The quantity in Column A is greater.

The two quantities are equal.

The two quantities are equal.

Explanation:

Recall for this question that the formulae for the area and circumference of a circle are, respectively:

For our two quantities, we have:

Quantity A

Therefore,

Taking the square root of both sides, we get:

Quantity B

Therefore,

Therefore, the two quantities are equal.

### Example Question #2 : How To Find The Length Of A Radius

If the diameter of a circle is equal to , then what is the value of the radius?

Explanation:

Given that the radius is equal to half the diameter, the value of the radius would be equal to  divided by 2. This gives us:

### Example Question #7 : How To Find The Length Of A Radius

The area of a circle is . Give its radius in terms of .

(Assume  is positive.)

Explanation:

The relation between the area of a circle  and its radius  is given by the formula

Since

:

We solve for :

Since  is positive, as is :

### Example Question #8 : How To Find The Length Of A Radius

The areas of five circles form an arithmetic sequence. The smallest circle has radius 4; the second smallest circle has radius 8. Give the radius of the largest circle.

Explanation:

The area of a circle with radius  is . Therefore, the areas of the circles with radii 4 and 8, respectively, are

and

The areas form an arithmetic sequence, the common difference of which is

.

The circles will have areas:

Since the area of the largest circle is , we can find the radius as follows:

The radius can be calculated now: