### All SAT II Math II Resources

## Example Questions

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

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:**

**Correct answer:**

The total surface area of the tetrahedron is 600 square centimeters; since the tetrahedron comprises four 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 #2 : 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 #3 : Faces, Face Area, And Vertices

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 #4 : Faces, Face Area, And Vertices

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 #5 : Faces, Face Area, And Vertices

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 #6 : 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 #7 : 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 #8 : Faces, Face Area, And Vertices

How many faces 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 #9 : Faces, Face Area, And Vertices

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.