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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

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

Sphere A has volume $972 \pi \textrm{ in}^{3}$ . Sphere B has surface area $400 \pi \textrm{ in}^{2}$ . Which is the greater quantity?

(a) The radius of Sphere A

(b) The radius of Sphere B

Possible Answers:

(a) is greater

It is impossible to tell from the information given

(a) and (b) are equal

(b) is greater

Correct answer:

(b) is greater

Explanation:

(a) Substitute $V = 972 \pi$ in the formula for the volume of a sphere:

$\frac{4}{3}\pi r^{3}= V$

$\frac{4}{3}\pi r^{3}= 972 \pi$

$\frac{4}{3}\pi r^{3} \div \frac{4}{3}\pi = 972 \pi \div \frac{4}{3}\pi$

$\frac{4}{3}\pi r^{3} \cdot \frac{3}{4\pi} = 972 \pi\cdot \frac{3}{4\pi}$

$r^{3} = 729$

$r =\sqrt[3]{ 729} = 9$ inches

(b) Substitute $A = 400 \pi$ in the formula for the surface area of a sphere:

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

$4\pi r^{2}= 400 \pi$

$4\pi r^{2} \div 4\pi = 400 \pi \div 4\pi$

$r^{2} = 100$

$r = \sqrt{100} = 10$ inches

(b) is greater.

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

is a positive number. Which is the greater quantity?

(A) The surface area of a sphere with radius

(B) The surface area of a cube with edges of length

Possible Answers:

(A) is greater

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

(A) and (B) are equal

(B) is greater

Correct answer:

(B) is greater

Explanation:

The surface area of a sphere is $4 \pi$ times the square of its radius, which here is ; the surface area of the sphere in (A) is $4 \pi t^{2}$ .

The area of one face of a cube is the square of the length of an edge, which here is , so the area of one face of the cube in (B) is $(2t) ^{2} = 4t^{2}$ . The cube has six faces so the total surface area is $6 \cdot 4t^{2} = 24t^{2}$ .

$\pi < 4$ , so $4 \pi t^{2} < 16 t^{2} < 24 t^{2}$ , giving the sphere less surface area. (B) is greater.

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Example Question #1 : How To Find The Volume Of A Cone

The height of Cone B is three times that of Cone A. The radius of the base of Cone B is one-half the radius of the base of Cone A.

Which is the greater quantity?

(a) The volume of Cone A

(b) The volume of Cone B

Possible Answers:

(b) is greater.

It is impossible to tell from the information given.

(a) is greater.

(a) and (b) are equal.

Correct answer:

(a) is greater.

Explanation:

Let r,h be the radius and height of Cone A, respectively. Then the radius and height of Cone B are $\frac{1}{2}r$ and , respectively.

(a) The volume of Cone A is $\frac{1}{3} \pi r^{2} h$ .

(b) The volume of Cone B is

$\frac{1}{3} \pi \left ( \frac{1}{2} r\right ) ^{2} \cdot 3 h = \frac{1}{3} \cdot \frac{1}{4} \cdot 3\cdot \pi r ^{2} h= \frac{1}{4} r ^{2} h$ .

Since $\frac{1}{4} r ^{2} h < \frac{1}{3} r ^{2} h$ , the cone in (a) has the greater volume.

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Example Question #1 : Volume Of A Cone

The volume of a cone whose height is three times the radius of its base is one cubic yard. Give its radius in inches.

Possible Answers:

$\frac{ 36 \sqrt[3]{ \pi^{2} }}{ \pi}$

$\frac{ \sqrt[3]{ \pi^{2} }}{ 3 \pi}$

$\frac{ \sqrt[3]{ \pi^{2} }}{ \pi}$

$\frac{ 12 \sqrt[3]{ \pi^{2} }}{ \pi}$

$\frac{ 3 \sqrt[3]{ \pi^{2} }}{ \pi}$

Correct answer:

$\frac{ 36 \sqrt[3]{ \pi^{2} }}{ \pi}$

Explanation:

The volume of a cone with base radius and height is

$V =\frac{1}{3} \pi r^{2}h$

The height is three times this, or . Therefore, the formula becomes

$V =\frac{1}{3} \pi r^{2} \cdot 3r$

$V = \pi r^{3}$

Set this volume equal to one and solve for :

$\pi r^{3} = 1$

$\pi r^{3} \div \pi = 1 \div \pi$

$r^{3}=\frac{ 1 }{ \pi}$

$r=\sqrt[3]{\frac{ 1 }{ \pi}} ={\frac{ \sqrt[3] {1} }{ \sqrt[3]{ \pi}}} ={\frac{ 1}{ \sqrt[3]{ \pi}}}={\frac{1 \cdot \sqrt[3]{ \pi^{2} }}{ \sqrt[3]{ \pi} \cdot \sqrt[3]{ \pi^{2}}} ={\frac{ \sqrt[3]{ \pi^{2} }}{ \pi}}$

This is the radius in yards; since the radius in inches is requested, multiply by 36.

$\frac{ \sqrt[3]{ \pi^{2} }}{ \pi} \times 36 = \frac{ 36 \sqrt[3]{ \pi^{2} }}{ \pi}$

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Example Question #1 : How To Find The Volume Of A Cone

The height of a given cylinder is one half the height of a given cone. The radii of their bases are equal.

Which of the following is the greater quantity?

(a) The volume of the cone

(b) The volume of the cylinder

Possible Answers:

(b) is the greater quantity

(a) is the greater quantity

It cannot be determined which of (a) and (b) is greater

(a) and (b) are equal

Correct answer:

(b) is the greater quantity

Explanation:

Call the radius of the base of the cone and the height of the cone. The cylinder will have bases of radius and height $\frac{1}{2} H$ .

In the formula for the volume of a cylinder, set r = R and $h = \frac{1}{2} H$ :

$V = \pi r^{2}h$

$V = \pi R ^{2} \cdot \frac{1}{2}H = \frac{1}{2} \pi R ^{2}H$

In the formula for the volume of a cone, set r = R and h = H :

$V = \frac{1}{3} \pi r^{2}h$

$V = \frac{1}{3} \pi R^{2}H$

$\frac{1}{2} > \frac{1}{3}$ , so

$\frac{1}{2} \pi R ^{2}H > \frac{1}{3} \pi R ^{2}H$ ,

meaning that the cylinder has the greater volume.

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Example Question #3 : How To Find The Volume Of A Cone

The radius of the base of a given cone is three times that of each base of a given cylinder. The heights of the cone and the cylinder are equal.

Which of the following is the greater quantity?

(a) The volume of the cone

(b) The volume of the cylinder

Possible Answers:

(a) is the greater quantity

It cannot be determined which of (a) and (b) is greater

(a) and (b) are equal

(b) is the greater quantity

Correct answer:

(a) is the greater quantity

Explanation:

If we let be the radius of each base of the cylinder, then is the radius of the base of the cone. We can let be their common height.

In the formula for the volume of a cylinder, set r = R and h = H :

$V = \pi r^{2}h$

$V = \pi R ^{2}H$

In the formula for the volume of a cone, set r = 3 R and h = H :

$V = \frac{1}{3} \pi r^{2}h$

$V = \frac{1}{3} \pi (3R)^{2}H = \frac{1}{3} \pi (9R^{2})H = 3\pi R ^{2}H$

3 > 1 , so $3\pi R ^{2}H > \pi R ^{2}H$ . The cone has the greater volume.

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Example Question #1 : How To Find The Volume Of A Cylinder

What is the volume of a cylinder with a radius of 6 meters and a height of 11 meters? Use 3.14 for $\pi$ .

Note: The formula for the volume of a cylinder is:

$V=\pi r^2h$

Possible Answers:

114.44m^3

207.24m^3

396m^3

1243.44 m^3

1432.23m^3

Correct answer:

1243.44 m^3

Explanation:

To calculate the volume, you must plug into the formula given in the problem. When you plug in, it should look like this: V=(3.14)(6^2)(11) . Multiply all of these out and you get 1243.44m^3 . The units are cubed because volume is always cubed.

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Example Question #1 : How To Find The Volume Of A Cylinder

The volume of a cylinder whose height is twice the diameter of its base is one cubic yard. Give its radius in inches.

Possible Answers:

$\frac{ 9 \sqrt[3]{ 2 \pi^{2} } } {\pi } \textrm{ in}$

$\frac{ 9 \sqrt[3]{ 2 \pi^{2} } } {4 \pi } \textrm{ in}$

$\frac{ 9 \sqrt[3]{ 2 \pi^{2} } } {2 \pi } \textrm{ in}$

$\frac{ \sqrt[3]{ 2 \pi^{2} } } {2\pi } \textrm{ in}$

$\frac{ 18 \sqrt[3]{ 2 \pi^{2} } } {\pi } \textrm{ in}$

Correct answer:

$\frac{ 18 \sqrt[3]{ 2 \pi^{2} } } {\pi } \textrm{ in}$

Explanation:

The volume of a cylinder with base radius and height is

$V = \pi r^{2}h$

The diameter of this circle is ; its height is twice this, or . Therefore, the formula becomes

$V = \pi r^{2} \cdot 4r = 4\pi r^{3}$

Set this volume equal to one and solve for :

$4\pi r^{3} = 1$

$4\pi r^{3} \div 4\pi = 1 \div 4\pi$

$r ^{3}= \frac{1}{4\pi }$

$r =\sqrt[3]{ \frac{1}{4\pi }} =\frac{ \sqrt[3]{ 1 }} { \sqrt[3]{ 4\pi }} =\frac{ \sqrt[3]{ 1 } \cdot \sqrt[3]{ 2 \pi^{2} } } { \sqrt[3]{ 4\pi } \cdot \sqrt[3]{ 2 \pi^{2} }}=\frac{ \sqrt[3]{ 2 \pi^{2} } } {\sqrt[3]{ 8\pi^{3} }}=\frac{ \sqrt[3]{ 2 \pi^{2} } } {2\pi }$

This is the radius in yards; multiply by 36 to get the radius in inches.

$\frac{ \sqrt[3]{ 2 \pi^{2} } } {2\pi } \cdot 36 = \frac{ 18 \sqrt[3]{ 2 \pi^{2} } } {\pi }$

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Example Question #3 : How To Find The Volume Of A Cylinder

What is the height of a cylinder with a volume of $646.875\pi$ in^3 and a radius of 7.5 ?

Possible Answers:

28125.5

11.5

86.25

56250

22.5

Correct answer:

11.5

Explanation:

Recall that the equation of for the volume of a cylinder is:

$V = \pi * r^2 * h$

For our values this is:

$646.875\pi = \pi * 7.5^2 * h= \pi * 56.25 * h$

Solve for :

h = 11.5

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Example Question #4 : How To Find The Volume Of A Cylinder

What is the volume of a cylinder with a height of in. and a radius of in?

Possible Answers:

$204\pi$ in^3

$300\pi$ in^3

$180\pi$ in^3

$720\pi$ in^3

$408\pi$ in^3

Correct answer:

$720\pi$ in^3

Explanation:

This is a rather direct question. Recall that the equation of for the volume of a cylinder is:

$V = \pi * r^2 * h$

For our values this is:

$A = \pi * 12^2 * 5=720\pi$

This is the volume of the cylinder.

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All ISEE Upper Level Quantitative Resources

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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 #3 : How To Find The Surface Area Of A Sphere

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

Example Question #1 : How To Find The Volume Of A Cone

Example Question #1 : Volume Of A Cone

Example Question #1 : How To Find The Volume Of A Cone

Example Question #3 : How To Find The Volume Of A Cone

Example Question #1 : How To Find The Volume Of A Cylinder

Example Question #1 : How To Find The Volume Of A Cylinder

Example Question #3 : How To Find The Volume Of A Cylinder

Example Question #4 : How To Find The Volume Of A Cylinder

All ISEE Upper Level Quantitative Resources

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