Polyhedron Properties and Equations

The video tutorial explores a problem involving a convex polyhedron with 12 faces, where 7 are triangles and 4 are quadrilaterals, leaving one unknown face. Using Euler's formula for connected planar graphs, the tutorial guides through calculating the number of edges and solving for the unknown face's sides. The solution reveals that the last face is a pentagon with 5 sides.

How many faces does the convex polyhedron have in total?

• 10

• 11

• 12

• 13

Tags

CCSS.6.G.A.4

How many vertices are present in the polyhedron?

• 9

• 10

• 11

• 12

What is Euler's formula for a connected planar graph?

• v - e + f = 2

• v + e + f = 2

• v + e - f = 2

• v - e - f = 2

Tags

CCSS.6.G.A.4

How many edges do the seven triangular faces contribute?

• 14

• 18

• 21

• 24

Tags

CCSS.6.G.A.4

What is the total number of edges contributed by the quadrilateral faces?

• 14

• 12

• 18

• 16

Tags

CCSS.3.OA.D.9

What must be true about the sum 37 + n for the number of edges to be a whole number?

• It must be prime

• It must be even

• It must be a multiple of 3

• It must be odd

Why must the unknown face have an odd number of edges?

• To ensure 37 + n is prime

• To ensure 37 + n is a multiple of 3

• To ensure 37 + n is odd

• To ensure 37 + n is even

Tags

CCSS.HSG.GPE.B.6

What is the final equation used to solve for n?

• 27 = (37 + n) / 2

• 21 = (37 + n) / 2

• 23 = (37 + n) / 2

• 25 = (37 + n) / 2

Tags

CCSS.1.G.A.1

CCSS.2.G.A.1

What is the number of sides of the unknown face?

• 6

• 5

• 4

• 3

Tags

CCSS.1.G.A.1

CCSS.2.G.A.1

What shape is the unknown face of the polyhedron?

• Quadrilateral

• Triangle

• Hexagon

• Pentagon

Tags

CCSS.1.G.A.1

CCSS.2.G.A.1