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ISEE Upper Level Quantitative : Circles

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

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All ISEE Upper Level Quantitative Resources

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Example Questions

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Example Question #3 : How To Find The Length Of A Radius

The tip of the minute hand of a giant clock has traveled $40 \pi$ 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:

(A) is greater

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

(B) is greater

(A) and (B) are equal

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,

$C \cdot 2 \frac{1}{2} = 40 \pi$

$C = 40 \pi \div 2 \frac{1}{2} = \frac{40 \pi}{1} \div \frac{5}{2} = \frac{40 \pi}{1} \cdot \frac{2}{5}= \frac{80 \pi}{5} = 16\pi$ feet.

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

$2 \pi r = C$

$2 \pi r = 16 \pi$

$r = 16 \pi \div 2 \pi = 8$

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

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Example Question #141 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Compare the two quantities:

Quantity A: The radius of a circle with area of $64\pi$

Quantity B: The radius of a circle with circumference of $50\pi$

Possible Answers:

The relationship cannot be determined from the information given.

The quantity in Column A is greater.

The two quantities are equal.

The quantity in Column B is greater.

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:

$A=\pi * r^2$

$C=2*\pi*r$

For our two quantities, we have:

Quantity A

$64\pi = \pi * r^2$

Therefore, r^2 = 64

Taking the square root of both sides, we get: r = 8

Quantity B

$50\pi = 2 * \pi * r$

Therefore, r = 25

Therefore, quantity B is greater.

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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 $144\pi$

Quantity B: The radius of a circle with circumference of $24\pi$

Possible Answers:

The relationship cannot be determined from the information given.

The two quantities are equal.

The quantity in Column B is greater.

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:

$A=\pi * r^2$

$C=2*\pi*r$

For our two quantities, we have:

Quantity A

$144\pi = \pi * r^2$

Therefore, r^2 = 144

Taking the square root of both sides, we get: r = 12

Quantity B

$24\pi = 2 * \pi * r$

Therefore, r = 12

Therefore, the two quantities are equal.

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

If the diameter of a circle is equal to $5x^{2}+4x-2$ , then what is the value of the radius?

Possible Answers:

$2.5x^{2}+2x-1$

$2(x^{2}+x)-1$

$2.5x^{2}+2x+1$

7x-1

Correct answer:

$2.5x^{2}+2x-1$

Explanation:

Given that the radius is equal to half the diameter, the value of the radius would be equal to $5x^{2}+4x-2$ divided by 2. This gives us:

$\frac{5x^{2}+4x-2}{2}$

$2.5x^{2}+2x-1$

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

The area of a circle is $81x^{2}$ . Give its radius in terms of .

(Assume is positive.)

Possible Answers:

$9 \pi x$

$\frac{9x}{\pi}$

$\frac{81x^{2}}{2\pi}$

$\frac{9x}{\sqrt{\pi}}$

$\frac{81x^{2}}{\pi}$

Correct answer:

$\frac{9x}{\sqrt{\pi}}$

Explanation:

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

$A = \pi r^{2}$

Since

$A= 81x^{2}$ :

$81x^{2} = \pi r^{2}$

We solve for :

$\pi r^{2} = 81x^{2}$

$\frac{\pi r^{2} }{\pi}= \frac{81x^{2}}{\pi}$

$r^{2} = \frac{81x^{2}}{\pi}$

$\sqrt{r^{2} }= \sqrt{\frac{81x^{2}}{\pi}}$

Since is positive, as is :

$r =\frac{ \sqrt{81} \cdot \sqrt{x^{2}}}{\sqrt{\pi}} = \frac{9x}{\sqrt{\pi}}$

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Example Question #1 : 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:

$4 \sqrt{13}$

$8\sqrt{13}$

Correct answer:

$4 \sqrt{13}$

Explanation:

The area of a circle with radius is $A = \pi r^{2}$ . Therefore, the areas of the circles with radii 4 and 8, respectively, are

$A_{1} = \pi \cdot 4^{2} = 16 \pi$

and

$A_{2} = \pi \cdot 8^{2} = 64\pi$

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

$64 \pi - 16 \pi = 48 \pi$ .

The circles will have areas:

$16 \pi, 64 \pi, 112 \pi, 160 \pi , 208 \pi$

Since the area of the largest circle is $208 \pi$ , we can find the radius as follows:

The radius can be calculated now:

$\pi r^{2} = 208 \pi$

$r^{2} = 208$

$r = \sqrt{208} = \sqrt{16} \cdot \sqrt{13} = 4 \sqrt{13}$

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Example Question #1 : Radius

Circle B has a radius $\frac{2}{3}$ as long as that of Circle A.

Which is the greater quantity?

(a) The area of Circle A

(b) Twice the area of Circle B

Possible Answers:

(b) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

(a) is greater.

Correct answer:

(a) is greater.

Explanation:

If we call the radius of Circle A , then the radius of Circle B is $\frac{2}{3} r$ .

The areas of the circles are:

(a) $\pi r^{2}$

(b) $\pi \left( \frac{2}{3} r \right ) ^{2} = \frac{4}{9} \pi r^{2}$

Twice the area of Circle B is

$2 \cdot \pi \left( \frac{2}{3} r \right ) ^{2} =2 \cdot \frac{4}{9} \pi r^{2} = \frac{8}{9} \pi r^{2} < \pi r^{2}$ ,

making (a) the greater number.

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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 #1 : 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 relationship cannot be determined from the information given.

The quantity in Column A is greater.

The two quantities are equal.

The quantity in Column B 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:

$2 \pi x - 10\pi$

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

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

$\pi x - 5\pi$

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

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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All ISEE Upper Level Quantitative Resources

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ISEE Upper Level Quantitative : Circles

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

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #3 : How To Find The Length Of A Radius

Example Question #141 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

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

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

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

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

Example Question #1 : Radius

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

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

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

All ISEE Upper Level Quantitative Resources

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