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

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

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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

Possible Answers:

(a) and (b) are equal.

(b) is greater.

(a) is greater.

It cannot be determined from the information given.

Correct answer:

(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?

Possible Answers:

 

Correct answer:

 

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 #3 : How To Find The Length Of A 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

Possible Answers:

(B) is greater

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

(A) and (B) are equal

(A) is greater

Correct answer:

(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 

Possible Answers:

The quantity in Column B is greater. 

The quantity in Column A is greater.

 

The two quantities are equal.

The relationship cannot be determined from the information given.

Correct answer:

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 

 

Possible Answers:

The quantity in Column B is greater. 

The relationship cannot be determined from the information given.

The two quantities are equal.

The quantity in Column A is greater.

Correct answer:

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 #6 : 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?

Possible Answers:

Correct answer:

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.)

Possible Answers:

Correct answer:

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.

Possible Answers:

Correct answer:

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:

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