SAT II Math II : Faces, Face Area, and Vertices

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

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Example Questions

Example Question #23 : 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:

Correct answer:

Explanation:

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 #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 #2 : 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:

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 #3 : Faces, Face Area, And Vertices

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 #4 : 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:

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 #5 : 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 #6 : 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 #7 : Faces, Face Area, And Vertices

How many faces 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 #8 : Faces, Face Area, And Vertices

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.

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