11.1 Notes.notebook
March 27, 2013
The tissue box is a rectangular solid. Let x = the number of corners, y = the number of flat surfaces, and z = the number of folded creases. What is an equation that relates x, y, and z for a rectangular solid?
Will your equation hold true for a cube?
A solid with a triangular top and bottom?
x + y = z + 2
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11.1 Space Figures and Cross Sections
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A polyhedron is a space figure, or three­dimensional figure, whose surfaces are polygons.
Each polygon is a face of the polyhedron. An edge is a segment that is formed by the intersection of two faces. A vertex is a point where three or more edges intersect.
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How many vertices, edges, and faces are in the polyhedron shown? List them.
Five vertices: D, E, F, G, H
Eight edges: DE, EF, GD, DH, EH, FH, and GH
Five faces: ΔDEH, ΔEFH, ΔFGH, ΔGDH, and quad. DEFG
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How many vertices, edges, and faces are in the polyhedron? List them.
6 vertices: R, S, T, U, V, W
9 edges: SR, ST, UR, UV, UT, RV, SW, VW, TW
5 faces: ΔURV, ΔSTW, quad. RSTU, quad. RSWV, quad. TWVU
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Is TV an edge? Explain why or why not.
No, an edge is the intersection of 2 faces, TV is contained in only one face.
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How many vertices, edges, and faces are in the polyhedron? List them.
Four vertices: R, S, T, U
Six edges: RS, RT, RU, ST, TU, and US
Four faces: ΔRST, ΔRTU, ΔRUS, ΔSTU
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Leonhard Euler, a Swiss mathematician, discovered a relationship among the numbers of face, vertices, and edges of any polyhedron. The result is known as Euler's Formula.
Euler's Formula
The sum of the number of faces (F) and vertices (V) of a polyhedron is two more than the number edges (E).
F + V = E + 2
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How many vertices, edges, and faces does the polyhedron have? Use your results to verify Euler's Formula.
F = 8
V = 12
E = 18
F + V = E + 2
8 + 12 = 18 + 2
20 = 20
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For the polyhedron, use Euler's Formula to find the missing number.
faces:
edges: 30
vertices: 20
12
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11.1 Notes.notebook
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For the polyhedron, use Euler's Formula to find the missing number.
faces: 20
edges: vertices: 12
30
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How many edges does the polyhedron have? Use Euler's Formula.
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In two dimensions, Euler's Formula reduces to F + V = E + 1, where F is the number of regions formed by V vertices linked by E segments.
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How can you verify Euler's Formula for a net for the solid shown.
Draw a net for the solid.
Number of regions: F = 8
Number of vertices: V = 22
Number of segments: E = 29
F + V = E + 1
8 + 22 = 29 + 1
30 = 30
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How can you verify Euler's Formula F + V = E + 2 for the solid?
6 + 8 = 12 + 2
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Draw a net for the solid.
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How can you verify Euler's Formula F + V = E + 1 for your two­dimensional net?
6 + 14 = 19 + 1
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How can you verify Euler's Formula for a net of a cube?
Draw a net for the cube.
There are 6 regions (F = 6), 14 vertices (V = 14), and 19 segments (E = 19). So, F + V = E + 1, or 6 + 14 = 19 + 1.
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How many faces, edges, and vertices are in the solid. List them.
5 faces: ABC, ACD, ADE, AEB, quad. BCDE
8 edges: AB, AC, ED, AE, BC, CD, DE, EB
5 vertices: A, B, C, D, E
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What is a net for the solid? Verify Euler's Formula for the net.
F + V = E + 1
5 + 8 = 12 + 1
13 = 13
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A cross section is the intersection of a solid and a plane.
You can think of a cross section as a very thin slice of the solid.
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What is the cross section formed by the plane and the solid shown?
The cross section is a rectangle.
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What is the cross section formed by a horizontal plane?
A circle
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What is the cross section formed by a vertical plane that divides the solid in half?
an isosc. trapezoid
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What is the cross section formed by the plane and the solid below?
Triangle
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To draw a cross section, you can sometimes use the idea from the postulate that describes the intersection of two planes as exactly one line.
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Draw a cross section formed by a vertical plane intersecting the front and right faces of the cube. What shape is the cross section?
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11.1 Notes.notebook
March 27, 2013
Draw a cross section formed by a vertical plane intersecting the front and right faces of the cube. What shape is the cross section?
Visualize a vertical plane intersecting the vertical faces in a parallel segments.
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March 27, 2013
Draw a cross section formed by a vertical plane intersecting the front and right faces of the cube. What shape is the cross section?
Visualize a vertical plane intersecting the vertical faces in a parallel segments.
Draw the parallel segments
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March 27, 2013
Draw a cross section formed by a vertical plane intersecting the front and right faces of the cube. What shape is the cross section?
Visualize a vertical plane intersecting the vertical faces in a parallel segments.
Draw the parallel segments
Join their endpoints.
Shade the cross section.
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Draw the cross section formed by a horizontal plane intersecting the left and right faces of the cube. What is the shape of the cross section?
square
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Draw a cross section formed by a vertical plane intersecting the right and left faces of the cube. What shape is the cross section?
a square
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What is the cross section formed by the cube and the plane containing the diagonals of a pair of opposite faces?
rectangle
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