Volume Day 1: Polyhedrons OBJECTIVE: To classify polyhedrons by their number of faces To determine and use Euler’s formula

Polyhedron a solid ( 3-D ) figure whose flat surfaces are polygons

Parts of a Polyhedron the flat surfaces of a polyhedron Faces Edge the segment where two faces intersect Vertex the point where three or more edges intersect

6 12 8 How many faces does this polyhedron have? Edges? Vertices? Cube

Polyhedrons are often classified (named) by the number of faces. 4 tetrahedron 5 pentahedron 6 hexahedron 7 heptahedron

# faces 8 octahedron 9 nonahedron 10 decahedron 11 undecahedron 12 dodecahedron

Regular Polyhedron a polyhedron whose faces are regular polygons and each face is congruent to the other faces and the faces meet each vertex in exactly the same way What shape is shown to the right? _____________________________ Regular octahedron (4 triangles on top & 4 triangles on bottom) *Only count outside surfaces as if it were solid

Discovering Euler’s Formula For each polyhedron, complete the table. Classify the polyhedron by the number of faces in column 1. Write the number of faces (flat surfaces) in column 2. Write the number of vertices (intersection points) in column 3. Write the number of edges (segments where two faces meet) in column 4. Count carefully! Pentahedron 5 5 8 Hexahedron 6 8 12 Heptahedron 7 10 15 Hexahedron 6 5 9 #4 *(3 triangles on top; 3 triangles on bottom)

Pentahedron 5 5 8 Hexahedron 6 8 12 Heptahedron 7 10 15 Hexahedron 6 5 9 Now that you have completed the table, let’s search the table to find a special relationship between the faces (F), vertices (V) and edges (E) in a polyhedron. (Work in your groups for a few minutes to attempt a formula.) *Write your formula in the space below. Now let’s see if you were correct…

Did you have one of these? This formula was first discovered by the famous Swiss mathematician Leonard Euler and is commonly know as Euler’s Formula. Did you have one of these? Using Euler’s Formula Euler’s Formula: (Can be written several ways.) F + V = E + 2 F + V – 2 = E OR F + V – E = 2 Now that you have discovered the formula relating the number of faces, vertices, and edges of a polyhedron, use it to answer each question below.

18 edges Euler’s Formula: F + V = E + 2 F + V – 2 = E F + V – E = 2 Ex 1 If a solid has 8 faces and 12 vertices, how many edges will it have? F + V = E + 2 F + V – 2 = E 8 + 12 = E + 2 8 + 12 – 2 = E 20 = E + 2 18= E 18 = E 18 edges

7 vertices Euler’s Formula: F + V = E + 2 F + V – 2 = E F + V – E = 2 Ex 2 If a solid has 7 faces and 12 edges, how many vertices will it have? F + V = E + 2 F + V – 2 = E 7 + V = 12 + 2 7 + V – 2 = 12 7 + V = 14 V + 5 = 12 V = 7 V = 7 7 vertices

ASSIGNMENT Volume Wkst. #1 Polyhedrons