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ISEE Upper Level Quantitative : Geometry

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

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All ISEE Upper Level Quantitative Resources

8 Diagnostic Tests 234 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

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

Circle 1 is inscribed inside a square. The square is inscribed inside Circle 2.

Which is the greater quantity?

(a) Twice the area of Circle 1

(b) The area of Circle 2

Possible Answers:

(a) and (b) are equal.

It is impossible to tell from the information given.

(a) is greater.

(b) is greater.

Correct answer:

(a) and (b) are equal.

Explanation:

If the radius of Circle 1 is , then the square will have sidelength equal to the diameter of the circle, or . Circle 2 will have as its diameter the length of a diagonal of the square, which by the $45 ^{\circ }-45 ^{\circ }-90^{\circ }$ Theorem is $\sqrt{2}$ times that, or $2r \sqrt{2}$ . The radius of Circle 2 will therefore be half that, or $r \sqrt{2}$ .

The area of Circle 1 will be $A = \pi r^{2}$ . The area of Circle 2 will be $A = \pi \left ( r\sqrt{2} \right ) ^{2} = \pi \left ( r^{2} \cdot 2 \right ) = 2 \pi r^{2}$ , twice that of Circle 1.

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

Compare the two quantities:

Quantity A: The area of a circle with radius

Quantity B: The circumference of a circle with radius

Possible Answers:

The quantity in Column B is greater.

The two quantities are equal.

The relationship cannot be determined from the information given.

The quantity in Column A is greater.

Correct answer:

The quantity in Column A is greater.

Explanation:

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

$A=\pi * r^2$

$C=2*\pi*r$

For our two quantities, we have:

Quantity A: $\pi * 13^2 = 169\pi$

Quantity B: $2 * \pi * 63 = 126\pi$

Therefore, quantity A is greater.

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

The radius of a circle is x - 5 . Give the area of the circle in terms of .

Possible Answers:

$\pi x^{2} -10 \pi x+ 25 \pi$

$4 \pi x^{2} -20 \pi x+ 50 \pi$

$\pi x - 5\pi$

$2 \pi x^{2} -20 \pi x+ 50 \pi$

$2 \pi x - 10\pi$

Correct answer:

$\pi x^{2} -10 \pi x+ 25 \pi$

Explanation:

The area of a circle with radius can be found using the formula

$A = \pi r^{2}$

Since r = x- 5 , the area is

$A = \pi (x- 5)^{2}$

$A = \pi (x^{2} - 2 \cdot x \cdot 5 + 5^{2})$

$A = \pi (x^{2} - 10 x + 25)$

$A = \pi x^{2} - 10 \pi x +25 \pi$

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Example Question #5 : How To Find The Area Of A Circle

The radius of a circle is x - 5 . Give the circumference of the circle in terms of .

Possible Answers:

$\pi x^{2} -10 \pi x+ 25 \pi$

$4 \pi x^{2} -20 \pi x+ 50 \pi$

$2 \pi x^{2} -20 \pi x+ 50 \pi$

$2 \pi x - 10\pi$

$\pi x - 5\pi$

Correct answer:

$2 \pi x - 10\pi$

Explanation:

The circumference of a circle is $2 \pi$ times its radius. Therefore, since the radius is x - 5 , the circumference is

$2 \pi \left (x - 5 \right ) = 2 \pi x - 2 \pi \cdot 5 = 2 \pi x - 10 \pi$

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

The areas of six circles form an arithmetic sequence. The second-smallest circle has a radius twice that of the smallest circle.

Which is the greater quantity?

(a) The area of the largest circle.

(b) Twice the area of the third-largest circle.

Possible Answers:

(b) is greater

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

(a) and (b) are equal

(a) is greater

Correct answer:

(b) is greater

Explanation:

Let be the radius of the smallest circle. Then the second-smallest circle has radius . Their areas, respectively, are

$A_{1} = \pi r ^{2}$

and

$A_{2} = \pi \left (2r \right )^{2} = 4 \pi r^{2}$

The areas form an arithmetic sequence, so their common difference is

$4 \pi r^{2} - \pi r^{2} = 3 \pi r^{2}$ .

The six areas are

$\pi r ^{2}, 4 \pi r ^{2}, 7 \pi r ^{2}, 10 \pi r ^{2}, 13 \pi r ^{2}, 17 \pi r ^{2}$

The third-largest circle has area $10 \pi r ^{2}$ ; twice this is $20 \pi r ^{2}$ . This is greater than the area of the largest circle, which is $17\pi r ^{2}$ . (b) is the greater quantity.

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Example Question #6 : How To Find The Area Of A Circle

The radii of six circles form an arithmetic sequence. The radius of the second-smallest circle is twice that of the smallest circle. Which of the following, if either, is the greater quantity?

(a) The area of the largest circle

(b) Twice the area of the third-smallest circle

Possible Answers:

(a) and (b) are equal

(a) is greater

(b) is greater

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

Correct answer:

(a) is greater

Explanation:

Call the radius of the smallest circle . The radius of the second-smallest circle is then , and the common difference of the radii is 2r - r = r .

The radii of the six circles are, from least to greatest:

r, 2r, 3r, 4r, 5r, 6r

The largest circle has area

$A_{6} = \pi (6r)^{2} = \pi \cdot 36 r^{2} = 36 \pi r^{2}$

The third-smallest circle has area:

$A_{3} = \pi (3r)^{2} = \pi \cdot 9 r^{2} = 9 \pi r^{2}$

Twice this is

$2 A_{3} = 2 \cdot 9 \pi r^{2} = 18 \pi r^{2}$

The area of the sixth circle is greater than twice that of the third-smallest circle, so the correct choice is that (a) is greater.

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Example Question #7 : How To Find The Area Of A Circle

Target

In the above figure, OA = AB = BC = CD .

Which is the greater quantity?

(a) Twice the area of inner gray ring

(b) The area of the white ring

Possible Answers:

(a) is the greater quantity

(b) is the greater quantity

(a) and (b) are equal

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

Correct answer:

(a) is the greater quantity

Explanation:

For the sake of simplicity, we will assume that OA = AB = BC = CD = 1 ; this reasoning is independent of the actual length.

The four concentric circles have radii 1, 2, 3, and 4, respectively, and their areas can be found by substituting each radius for in the formula $A = \pi r^{2}$ :

$A _{1}= \pi r^{2} = \pi \cdot 1 ^{2} = \pi$

$A _{2}= \pi r^{2} = \pi \cdot 2^{2} =4 \pi$

$A _{3}= \pi r^{2} = \pi \cdot 3^{2} =9 \pi$

$A _{4}= \pi r^{2} = \pi \cdot 4^{2} =16 \pi$

The white ring has as its area the difference of the areas of the second-largest and third-largest circles:

$A_{3} - A_{2} = 9 \pi - 4 \pi = 5 \pi$

The inner gray ring has as its area the difference of the areas of the third-largest and smallest circles:

$A_{2} - A_{1} = 4 \pi - \pi = 3 \pi$ .

Twice this is $2 \cdot 3 \pi = 6 \pi$ , which is greater than the area of the white ring.

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

Target

In the above figure, OA = AB = BC = CD .

Which is the greater quantity?

(a) Six times the area of the white circle

(b) The area of the outer ring

Possible Answers:

(b) is the greater quantity

(a) is the greater quantity

(a) and (b) are equal

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

Correct answer:

(b) is the greater quantity

Explanation:

For the sake of simplicity, we will assume that OA = AB = BC = CD = 1 ; this reasoning is independent of the actual length.

The four concentric circles have radii 1, 2, 3, and 4, respectively, and their areas can be found by substituting each radius for in the formula $A = \pi r^{2}$ :

$A _{1}= \pi r^{2} = \pi \cdot 1 ^{2} = \pi$

$A _{2}= \pi r^{2} = \pi \cdot 2^{2} =4 \pi$

$A _{3}= \pi r^{2} = \pi \cdot 3^{2} =9 \pi$

$A _{4}= \pi r^{2} = \pi \cdot 4^{2} =16 \pi$

The outer gray ring is the region between the largest and second-largest circles, and has area

$A _{4} - A _{3} = 16 \pi - 9 \pi = 7 \pi$

Six times the area of the white (inner) circle is $6 A_{1} = 6 \pi$ , which is less than the area of the outer ring, $7 \pi$ .

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Example Question #1 : How To Find Circumference

A giant clock has a minute hand five yards in length. Since noon, the tip of the minute hand has traveled $200 \pi$ feet. Which is the greater quantity?

(A) The amount of time that has passed since noon

(B) The amount of time until midnight

Possible Answers:

(A) is greater

(B) is greater

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

(A) and (B) are equal

Correct answer:

(A) is greater

Explanation:

Five yards is equal to fifteen feet, which is the length of the minute hand. Subsequently, fifteen feet is the radius of the circle traveled by its tip in one hour; the circumference of this circle is $2 \pi$ times this, or

$C = 2 \pi \cdot 15 = 30 \pi$ feet.

In one six-hour period, the minute hand revolves six times, so its tip travles six times the circumference, or

$30 \pi \times 6 = 180 \pi$

The clock has traveled farther than this, so the time is later than 6:00 PM, and more time has elapsed since noon than is left until midnight. This makes (A) greater.

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Example Question #2 : How To Find Circumference

Compare the two quantities:

Quantity A: The area of a circle with radius

Quantity B: The circumference of a circle with radius 8.1

Possible Answers:

The two quantities are equal.

The quantity in Column B is greater.

The relationship cannot be determined from the information given.

The quantity in Column A is greater.

Correct answer:

The quantity in Column A is greater.

Explanation:

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

$A=\pi * r^2$

$C=2*\pi*r$

For our two quantities, we have:

Quantity A: $\pi * 5^2 = 25\pi$

Quantity B: $2 * \pi * 8.1 = 16.2\pi$

Therefore, quantity A is greater.

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All ISEE Upper Level Quantitative Resources

8 Diagnostic Tests 234 Practice Tests Question of the Day Flashcards Learn by Concept

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ISEE Upper Level Quantitative : Geometry

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

All ISEE Upper Level Quantitative Resources

Example Questions

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

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

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

Example Question #5 : How To Find The Area Of A Circle

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

Example Question #6 : How To Find The Area Of A Circle

Example Question #7 : How To Find The Area Of A Circle

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

Example Question #1 : How To Find Circumference

Example Question #2 : How To Find Circumference

All ISEE Upper Level Quantitative Resources

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