SAT II Math II : 3-Dimensional Geometry

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

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Example Questions

Example Question #1 : Faces, Face Area, And Vertices

A regular octahedron has eight congruent faces, each of which is an equilateral triangle. 

The total surface area of a given regular octahedron is 400 square centimeters. To the nearest tenth of a centimeter, what is the length of each edge?

Possible Answers:

Correct answer:

Explanation:

The total surface area of the octahedron is 400 square centimeters; since the octahedron comprises eight congruent faces, each has area  square centimeters.

The area of an equilateral triangle is given by the formula

Set  and solve for 

 centimeters.

 

Example Question #411 : Sat Subject Test In Math Ii

A regular icosahedron has twenty congruent faces, each of which is an equilateral triangle. 

A given regular icosahedron has edges of length four inches. Give the total surface area of the icosahedron.

Possible Answers:

Correct answer:

Explanation:

The area of an equilateral triangle is given by the formula

Since there are twenty equilateral triangles that comprise the surface of the icosahedron, the total surface area is 

Substitute :

  square inches.

Example Question #111 : Geometry

How many faces does a polyhedron with nine vertices and sixteen edges have?

Possible Answers:

Correct answer:

Explanation:

By Euler's Formula, the relationship between the number of vertices , the number of faces , and the number of edges  of a polyhedron is 

Set  and  and solve for :

The polyhedron has nine faces.

Example Question #31 : 3 Dimensional Geometry

How many edges does a polyhedron with eight vertices and twelve faces have?

Possible Answers:

Insufficient information is given to answer the question. 

Correct answer:

Explanation:

By Euler's Formula, the relationship between the number of vertices , the number of faces , and the number of edges  of a polyhedron is 

Set  and  and solve for :

The polyhedron has eighteen edges.

Example Question #2 : Faces, Face Area, And Vertices

How many faces does a polyhedron with ten vertices and fifteen edges have?

Possible Answers:

Insufficient information is given to answer the question.

Correct answer:

Explanation:

By Euler's Formula, the relationship between the number of vertices , the number of faces , and the number of edges  of a polyhedron is 

Set  and  and solve for :

The polyhedron has seven faces.

Example Question #3 : Faces, Face Area, And Vertices

How many faces does a polyhedron with ten vertices and sixteen edges have?

Possible Answers:

Correct answer:

Explanation:

By Euler's Formula, the relationship between the number of vertices , the number of faces , and the number of edges  of a polyhedron is 

Set  and  and solve for :

The polyhedron has eight faces.

Example Question #32 : 3 Dimensional Geometry

A convex polyhedron with eighteen faces and forty edges has how many vertices?

Possible Answers:

Correct answer:

Explanation:

The number of vertices, edges, and faces of a convex polygon——are related by the Euler's formula:

Therefore, set  and solve for :

The polyhedron has twenty-four faces.

Example Question #4 : Faces, Face Area, And Vertices

How many edges does a polyhedron with fourteen vertices and five faces have?

Possible Answers:

Correct answer:

Explanation:

By Euler's Formula, the relationship between the number of vertices , the number of faces , and the number of edges  of a polyhedron is 

.

Set  and  and solve for :

The polyhedron has seventeen edges.

Example Question #1 : 3 Dimensional Axes And Coordinates

Which of the following numbers comes closest to the length of line segment in three-dimensional coordinate space whose endpoints are the origin and the point  ?

Possible Answers:

Correct answer:

Explanation:

Use the three-dimensional version of the distance formula:

The closest of the five choices is 7.

Example Question #1 : 3 Dimensional Axes And Coordinates

A line segment  in three-dimensional space has midpoint  has midpoint .

 has Cartesian coordinates  has Cartesian coordinates . Give the -coordinate of .

Possible Answers:

Correct answer:

Explanation:

The midpoint formula for the -coordinate

will be applied twice, once to find the -coordinate of , then again to find that of .

 

First, set , the -coordinate of , and , the -coordinate of , and solve for , the -coordinate of :

 

Now, set , the  -coordinate of , and , the -coordinate of , and solve for , the -coordinate of :

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