Faces, Face Area, and Vertices - SAT Subject Test in Math II
Card 0 of 9
How many edges does a polyhedron with eight vertices and twelve faces have?
How many edges does a polyhedron with eight vertices and twelve faces have?
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.
By Euler's Formula, the relationship between the number of vertices
Set
The polyhedron has eighteen edges.
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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?
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?
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.
The total surface area of the tetrahedron is 600 square centimeters; since the tetrahedron comprises four congruent faces, each has area
The area of an equilateral triangle is given by the formula
Set
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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?
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?
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.
The total surface area of the octahedron is 400 square centimeters; since the octahedron comprises eight congruent faces, each has area
The area of an equilateral triangle is given by the formula
Set
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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.
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.
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.
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
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How many edges does a polyhedron with fourteen vertices and five faces have?
How many edges does a polyhedron with fourteen vertices and five faces have?
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.
By Euler's Formula, the relationship between the number of vertices
Set
The polyhedron has seventeen edges.
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How many faces does a polyhedron with nine vertices and sixteen edges have?
How many faces does a polyhedron with nine vertices and sixteen edges have?
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.
By Euler's Formula, the relationship between the number of vertices
Set
The polyhedron has nine faces.
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How many faces does a polyhedron with ten vertices and fifteen edges have?
How many faces does a polyhedron with ten vertices and fifteen edges have?
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.
By Euler's Formula, the relationship between the number of vertices
Set
The polyhedron has seven faces.
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How many faces does a polyhedron with ten vertices and sixteen edges have?
How many faces does a polyhedron with ten vertices and sixteen edges have?
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.
By Euler's Formula, the relationship between the number of vertices
Set
The polyhedron has eight faces.
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A convex polyhedron with eighteen faces and forty edges has how many vertices?
A convex polyhedron with eighteen faces and forty edges has how many vertices?
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.
The number of vertices, edges, and faces of a convex polygon—
Therefore, set
The polyhedron has twenty-four faces.
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