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SAT II Math II : 3-Dimensional Geometry

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Example Questions

Example Question #21 : 3 Dimensional Geometry

Find the volume of a sphere with a surface area of $36\pi$ .

Possible Answers:

$18\pi$

$27\pi$

$36\pi$

$96\pi$

$72\pi$

Correct answer:

$36\pi$

Explanation:

Write the surface area formula for a sphere.

$A= 4\pi r^2$

Substitute the area.

$36\pi= 4\pi r^2$

Divide both sides by $4\pi$ .

$\frac{36\pi}{4\pi}=\frac{ 4\pi r^2}{4\pi}$

r^2 =9

r=3

Write the formula for the volume of a sphere.

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

Substitute the radius.

$V=\frac{4}{3}\pi (3)^3 = 36\pi$

The answer is: $36\pi$

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Example Question #1 : Surface Area

A rectangular swimming pool is meters deep throughout and meters wide. Its length is ten meters greater than twice its width. Which of the following expressions gives the total surface area, in square meters, of the inside of the pool?

Possible Answers:

$4W^{2} + 3DW + 10W+ 20D$

$4W^{2} + 6DW + 20W+ 20D$

$2W^{2} + 6DW + 10W+ 20D$

$2DW^{2}+ 10DW$

$2W^{2} + 3DW + 10W+ 10D$

Correct answer:

$2W^{2} + 6DW + 10W+ 20D$

Explanation:

Since the length of the pool is ten meters longer than twice its width , its length is 2W + 10 .

The inside of the pool can be seen as a rectangular prism. The bottom, or the base, has dimensions and 2W + 10 , so its area is the product of these:

$W (2W + 10) = 2W^{2}+ 10W$

The sides of the pool have depth . Two sides have width and therefore have area .

Two sides have width 2W + 10 and therefore have area

$D \left (2W + 10 \right ) = 2DW + 10D$

The total area of the inside of the pool is

$2W^{2}+ 10W + 2 (DW ) + 2 (2DW + 10D)$

$=2W^{2}+ 10W + 2 DW + 4DW + 20D$

$=2W^{2} + 6DW + 10W+ 20D$

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Example Question #2 : Surface Area

A water tank takes the shape of a closed rectangular prism whose exterior has height 30 feet, length 20 feet, and width 15 feet. Its walls are one foot thick throughout. What is the total surface area of the interior of the tank?

Possible Answers:

$2,204 \textrm{ ft}^{2}$

$1,223 \textrm{ ft}^{2}$

$1,102\textrm{ ft}^{2}$

$2,446\textup{ ft}^{2}$

$2,694\textup{ ft}^{2}$

Correct answer:

$2,204 \textrm{ ft}^{2}$

Explanation:

The height, length, and width of the interior tank are each two feet less than the corresponding dimension of the exterior of the tank, so the dimensions of the interior are 28, 18, and 13 feet. The surface area of the interior is what we are looking for here. It comprises six rectangles:

Two with area $28 \times 18 = 504$ square feet;

Two with area $28 \times 13 = 364$ square feet;

Two with area $18 \times 13 = 234$ square feet.

Add:

$2 \times 504 + 2 \times 364 + 2 \times 234 = 1,008 + 728 + 468 = 2,204$ square feet.

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Example Question #1 : Surface Area

What is the surface area of a cube with a side length of 5?

Possible Answers:

150

125

Correct answer:

150

Explanation:

If you were to take apart a cube so that you could lay it flat on a surface, you would be able to see that a cube is just made up of 6 identical squares. The area of one square is the length of the side squared, so the surface area of the cube would be denoted with the formula:

A=6*s^2

In this case the side length is 5, so plugging that into the formula will get the answer.

A=6*(5)^2=6*25=150

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Example Question #3 : Surface Area

Find the surface area of a rectangular prism with length, width, and height dimensions of $\frac{1}{2}$ , $\frac{3}{4}$ , and $\frac{5}{2}$ , respectively.

Possible Answers:

$\frac{15}{8}$

$\frac{15}{16}$

Correct answer:

Explanation:

Be careful not to confuse surface area with volume!

There are 6 faces in a rectangular prism, and we will need the sum of all the areas of each face.

Write the formula.

A=2(LW+LH+WH)

Substitute the dimensions.

$A= 2[(\frac{1}{2})(\frac{3}{4})+(\frac{1}{2})(\frac{5}{2})+(\frac{3}{4})(\frac{5}{2})]$

Evaluate each product in the bracket.

$A= 2[\frac{3}{8}+\frac{5}{4}+\frac{15}{8}] = \frac{3}{4}+\frac{10}{4}+\frac{15}{4}$

Combine the terms of the numerator.

$A = \frac{28}{4} =7$

The answer is:

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Example Question #101 : Geometry

What is the surface area of a cube with a side length of ?

Possible Answers:

54x^2

54x^6

27x^3

27x^2

Correct answer:

54x^2

Explanation:

Write the formula for the surface area of a cube.

A= 6s^2

Substitute the side length.

A= 6(3x)^2 = 6(3x)(3x) = 54x^2

The answer is: 54x^2

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Example Question #1 : Surface Area

Find the surface area of a sphere with a radius of 3.

Possible Answers:

$18\pi$

$36\pi$

$72\pi$

$48\pi$

$27\pi$

Correct answer:

$36\pi$

Explanation:

Write the formula for the surface area of a sphere.

$A=4\pi r^2$

Substitute the radius into the equation.

$A=4\pi(3)^2 = 4\pi (9) = 36\pi$

The answer is: $36\pi$

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Example Question #2 : Surface Area

Find the surface area of a sphere with a radius of 2.

Possible Answers:

$\frac{16}{3}\pi$

$8\pi$

$16\pi$

$4\pi$

$\frac{8}{3}\pi$

Correct answer:

$16\pi$

Explanation:

Write the formula for the surface area of a sphere.

$A=4\pi r^2$

Substitute the radius.

$A=4\pi (2)^2 = 16\pi$

The answer is: $16\pi$

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Example Question #1 : Surface Area

Find the surface area of a cube if the area of one of the square faces is .

Possible Answers:

$\textup{The answer cannot be determined.}$

256

Correct answer:

Explanation:

Write the formula for the surface area of a cube.

A = 6s^2

The area of a square side is already given. Substitute that as the s^2 term.

A=6(16) = 96

The answer is:

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Example Question #21 : 3 Dimensional Geometry

A regular tetrahedron has four congruent faces, each of which is an equilateral triangle.

The total surface area of a given regular tetrahedron is 600 square centimeters. To the nearest tenth of a centimeter, what is the length of each edge?

Possible Answers:

$12.2 \textrm{ cm}$

$11.9 \textrm{ cm}$

$24.5 \textrm{ cm}$

$37.2\textrm{ cm}$

$18.6 \textrm{ cm}$

Correct answer:

$18.6 \textrm{ cm}$

Explanation:

The total surface area of the tetrahedron is 600 square centimeters; since the tetrahedron comprises four congruent faces, each has area $600 \div 4 = 150$ square centimeters.

The area of an equilateral triangle is given by the formula

$A = \frac{s^{2}\sqrt{3}}{4}$

Set A = 150 and solve for :

$\frac{s^{2}\sqrt{3}}{4} = 150$

$\frac{s^{2}\sqrt{3}}{4} \cdot \frac{4}{\sqrt{3}}=1 50 \cdot \frac{4}{\sqrt{3}}$

$s^{2} = \frac{600}{ \sqrt{3}} \approx \frac{600}{1.732} \approx 346.41$

$s \approx \sqrt{346.41} \approx 18.6$ centimeters.

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SAT II Math II : 3-Dimensional Geometry

Study concepts, example questions & explanations for SAT II Math II

All SAT II Math II Resources

Example Questions

Example Question #21 : 3 Dimensional Geometry

Example Question #1 : Surface Area

Example Question #2 : Surface Area

Example Question #1 : Surface Area

Example Question #3 : Surface Area

Example Question #101 : Geometry

Example Question #1 : Surface Area

Example Question #2 : Surface Area

Example Question #1 : Surface Area

Example Question #21 : 3 Dimensional Geometry

All SAT II Math II Resources

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