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SAT II Math I : Geometry

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Example Questions

Example Question #22 : Volume

Find the volume of a sphere with the diameter of .

Possible Answers:

$12\pi$

$9\pi$

$27\pi$

$\frac{9}{4}\pi$

$\frac{9}{2}\pi$

Correct answer:

$\frac{9}{2}\pi$

Explanation:

Write the equation for the volume of a sphere.

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

The radius is half the diameter, or $\frac{3}{2}$ .

Substitute the radius into the equation.

$V=\frac{4}{3}\pi( \frac{3}{2})^3$

$V=\frac{4}{3}\pi( \frac{27}{8})$

Reduce the fractions.

The answer is: $\frac{9}{2}\pi$

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Example Question #22 : 3 Dimensional Geometry

Determine the volume of a rectangular prism if the length is $\frac{1}{2}$ , width is $\frac{2}{3}$ , and the height is $\frac{5}{6}$ .

Possible Answers:

$\frac{1}{3}$

$\frac{2}{3}$

$\frac{9}{5}$

$\frac{5}{18}$

$\frac{5}{9}$

Correct answer:

$\frac{5}{18}$

Explanation:

Write the formula for the volume of a rectangular prism.

$A=\textup{ (Length) (Width) (Height)}$

Substitute the dimensions into the formula.

$A = \frac{1}{2}\cdot \frac{2}{3}\cdot \frac{5}{6} = \frac{10}{36}$

Reduce this fraction.

$\frac{5\times 2}{18\times 2}$

The answer is: $\frac{5}{18}$

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Example Question #24 : Volume

The length of a box is half its height and two-thirds its width. The volume of the box is four cubic meters. Give the length of the box to the nearest centimeter.

Possible Answers:

$144 \textrm{ cm}$

None of these

$229\textrm{ cm}$

$175 \textrm{ cm}$

$110 \textrm{ cm}$

Correct answer:

$110 \textrm{ cm}$

Explanation:

Call , , and the length, width, and height of the crate.

The length of the crate is half its height, so

$L = \frac{1}{2} H$

$2 \cdot L = 2 \cdot \frac{1}{2} H$

2 L = H

The length of the crate is two-thirds its width, so

$L = \frac{2}{3} W$

$\frac{3} {2} \cdot L = \frac{3} {2} \cdot \frac{2}{3} W$

$\frac{3} {2} L = W$

The dimensions of the crate in terms of are , $\frac{3} {2} L$ , and . The volume is their product:

V= LWH

Substitute:

LWH = V

$L \cdot \frac{3} {2} L \cdot 2L = 4$

$3 L ^{3} = 4$

Solve for :

$3 L ^{3} = 4$

$\frac{3 L ^{3} }{3} = \frac{4 }{3}$

$L ^{3} = \frac{4 }{3}$

$L =\sqrt[3]{ \frac{4 }{3}}$

$L \approx 1.101$ meters.

Since one meter comprises 100 centimeters, multiply by 100 to convert to centimeters:

$1.101 \cdot 100 = 110.1$ ,

which rounds to 110 centimeters.

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Example Question #21 : 3 Dimensional Geometry

A circular swimming pool has diameter 32 meters and depth meters throughout. Which of the following expressions gives the total area of the inside of the pool, in square meters?

Possible Answers:

None of the other responses is correct.

$256\pi + 16 \pi t$

$512\pi + 32 \pi t$

$256\pi + 32 \pi t$

$512\pi + 16 \pi t$

Correct answer:

$256\pi + 32 \pi t$

Explanation:

The bottom of the pool is a circle with diameter 32, and, subsequently, radius half this, or 16; its area is

$B = \pi r^{2} = \pi \times 16^{2} = 256 \pi$

The side of the pool is the lateral surface of a cylinder with radius 16 and height ; the area of this is

$LA = 2 \pi r h = 2 \pi \cdot 16 \cdot t = 32 \pi t$

The area of the inside of the pool is the sum of these two, or

$A = 256 \pi + 32 \pi t$

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Example Question #1 : Surface Area

A circular swimming pool at an apartment complex has diameter 50 feet and depth six feet throughout.

The apartment manager needs to get the interior of the swimming pool painted. The paint she wants to use covers 350 square feet per gallon. How many one-gallon cans of paint will she need to purchase?

Possible Answers:

The correct answer is not given among the other responses.

Correct answer:

Explanation:

The pool can be seen as a cylinder with depth (or height) six feet and a base with diameter 50 feet - and radius half this, or 25 feet.

The bottom of the pool - the base of the cylinder - is a circle with radius 25 feet, so its area is

$B = \pi r ^{2} = 3.14 \cdot 25 \cdot 25 = 1,962.5$ square feet.

Its side - the lateral face of the cylinder - has area

$LA = 2 \pi r h = 2 \cdot 3.14 \cdot 25 \cdot 6 = 942$ square feet.

Their sum - the total area to be painted - is 1,962.5 + 942 = 2,904.5 square feet. Since one gallon of paint covers 350 square feet, divide:

$2,904.5 \div 350 \approx 8.3$

Eight cans of paint and part of a ninth will be required, so the correct response is nine.

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Example Question #1 : Surface Area

Example cylinder

Figure not drawn to scale.

Find the surface area of the cylinder above.

Possible Answers:

87.25 in²

122.14 in²

94.25 in²

73.44 in²

56.55 in²

Correct answer:

94.25 in²

Explanation:

In order to find the surface area of a cylinder, you need to find the surface areas of the circles that are the top and bottom of the cylinder (2 x pi x radius²) and add it to the surface are of the rectangle that is the side of the cylinder (diameter x height).

$\small \small \small \small Surface Area=(2\pi r{^2})+(\pi2r\times h)$

$\small \small (2\pi\times 3{^2})+(6\pi\times 2)$

$\small \small (6.28\times 9)+(18.84\times 2)$

$\small 56.54 + 37.70=94.25$

The surface area of the cylinder is 94.25 in²

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Example Question #1 : Surface Area

Example images

Figure not drawn to scale

What is the surface area of the sphere above?

Possible Answers:

712.12 yd²

512.63 yd²

815.44 yd²

615.75 yd²

615.75 yd³

Correct answer:

615.75 yd²

Explanation:

In order to find the surface area of a sphere, you must use the equation below:

$\small Surface Area=4\pi r{^{2}}$

$\small 4\pi(7{^{2}})=4\pi (49)=196\pi=615.75$

The surface area of the sphere is 615.75 yd²

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Example Question #1 : Advanced Geometry

What is the surface area of the following cone?

Possible Answers:

$\pi$ m^2

Correct answer:

$\pi$ m^2

Explanation:

The formula for the surface area of a cone is:

A = $\pi$ $\pi$ ,

where represents the radius of the cone base and represents the slant height of the cone.

Plugging in our values, we get:

A = $\pi$ $\pi$

A = $\pi$ m^2

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Example Question #1 : Cones

The surface area of cone $\small E$ is $\small 70\pi$ . If the radius of the base of the cone is , what is the height of the cone?

Possible Answers:

$\small \small 2\sqrt{14}$

$\small 2\sqrt{5}$

$\small 45$

$\small 3$

$\small 9$

Correct answer:

$\small \small 2\sqrt{14}$

Explanation:

To figure out $\small h$ , we must use the equation for the surface area of a cone, $\small A=\pi r^2+\pi rs$ , where $\small r$ is the radius of the base of the cone and $\small s$ is the length of the diagonal from the tip of the cone to any point on the base's circumference. We therefore first need to solve for $\small s$ by plugging what we know into the equation:

$\small 70\pi=(5^2)\pi +5s\pi=25\pi +5s\pi$

This equation can be reduced to:

$\small 45\pi=5s\pi$

$\small 9=s$

For a normal right angle cone, $\small s$ represents the line from the tip of the cone running along the outside of the cone to a point on the base's circumference. This line represents the hypotenuse of the right triangle formed by the radius and height of the cone. We can therefore solve for $\small h$ using the Pythagorean theorem:

$\small s^2 = r^2+h^2$

$\small h=\sqrt{s^2-r^2}$

Our $\small h$ is therefore:

$\small h=\sqrt{9^2-5^2}=\sqrt{81-25}=\sqrt{56}=2\sqrt{14}$

The height of cone $\small E$ is therefore $\small 2\sqrt{14}$

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Example Question #2 : Advanced Geometry

A circle of radius five is cut into two pieces, $30\%$ and $70\%$ . The larger section is thrown away. The smaller section is curled until the two straight edges meet, and a bottom is made for the cone.

What is the area of the bottom?

Possible Answers:

$\frac{9\pi }{4}$

$2\pi$

$25\pi$

$3\pi$

Correct answer:

$\frac{9\pi }{4}$

Explanation:

When the smaller portion of the circle is curled in, it will make the top of a cone. The circumfrence of the circle on the bottom is $2\pi r$ (where r is the radius of the circle on the bottom). The circumference of the bottom is also $30\%$ of the circumfrence of the original larger circle, which is $.3C=.3(2\pi R)=.3(10\pi)=3\pi$ (where R is the radius of the original, larger circle)

Therefore we use the circumference formula to solve for our new r:

$2\pi r=3\pi, r=\frac{3}{2}$

Substituting this value into the area formula, the area of the small circle becomes:

$A=\pi r^2=\frac{9\pi}{4}$

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SAT II Math I : Geometry

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

All SAT II Math I Resources

Example Questions

Example Question #22 : Volume

Example Question #22 : 3 Dimensional Geometry

Example Question #24 : Volume

Example Question #21 : 3 Dimensional Geometry

Example Question #1 : Surface Area

Example Question #1 : Surface Area

The surface area of the cylinder is 94.25 in²

Example Question #1 : Surface Area

What is the surface area of the sphere above?

The surface area of the sphere is 615.75 yd²

Example Question #1 : Advanced Geometry

Example Question #1 : Cones

Example Question #2 : Advanced Geometry

All SAT II Math I Resources

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HIGH SCHOOL

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K-8

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math tutoring

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SAT II Math I : Geometry

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

All SAT II Math I Resources

Example Questions

Example Question #22 : Volume

Example Question #22 : 3 Dimensional Geometry

Example Question #24 : Volume

Example Question #21 : 3 Dimensional Geometry

Example Question #1 : Surface Area

Example Question #1 : Surface Area

The surface area of the cylinder is 94.25 in2

Example Question #1 : Surface Area

What is the surface area of the sphere above?

The surface area of the sphere is 615.75 yd2

Example Question #1 : Advanced Geometry

Example Question #1 : Cones

Example Question #2 : Advanced Geometry

All SAT II Math I Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

The surface area of the cylinder is 94.25 in²

The surface area of the sphere is 615.75 yd²