### All SAT II Math II Resources

## 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:**

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:**

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:**

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:**

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:**

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:**

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:**

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:**

.

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:**

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:**

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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