In two dimensions, Euler's Formula reduces to
Problem 3 Verifying Euler's Formula in Two Dimensions
How can you verify Euler's Formula for a net for the solid in Problem 2?
Plan
What do you use for the variables?
| In 3-D | In 2-D | |
| F: Faces | → | Regions |
| V: Vertices | → | Vertices |
| E: Edges | → | Segments |
Draw a net for the solid.
Number of regions:
Number of vertices:
Number of segments:
✓ Got It?
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Use the solid below.
- How can you verify Euler's Formula
- Draw a net for the solid.
- How can you verify Euler's Formula
- How can you verify Euler's Formula
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.
Problem 4 Describing a Cross Section
Think
How can you see the cross section?
Mentally rotate the solid so that the plane is parallel to your face.
What is the cross section formed by the plane and the solid below?
The cross section is a rectangle.
✓ Got It?
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For the solid below, what is the cross section formed by each of the following planes?
- a horizontal plane
- a vertical plane that divides the solid in half
Table of Contents
- 6-1 The Polygon Angle-Sum Theorems
- 6-2 Properties of Parallelograms
- 6-3 Proving That a Quadrilateral Is a Parallelogram
- 6-4 Properties of Rhombuses, Rectangles, and Squares
- 6-5 Conditions for Rhombuses, Rectangles, and Squares
- 6-6 Trapezoids and Kites
- 6-7 Polygons in the Coordinate Plane
- 6-8 and 6-9 Coordinate Geometry and Coordinate Proofs