10.1.3 Formulas in Three Dimensions Key Objectives • Apply Euler’s Formula to find the number of vertices, edges, and faces of a polyhedron. • Develop and apply the distance and midpoint formulas in three dimensions. Key Terms • A polyhedron is formed by four or more polygons that intersect only at their edges. • Space is the set of all points in three dimensions. Formulas • Euler’s Formula For any polyhedron with V vertices, E edges, and F faces, V − E + F = 2. • Diagonal of a Right Rectangular Prism The length of a diagonal of a right rectangular prism with length l , width w , and height h is d l w h 2 2 2 = + + . • Distance Formula in Three Dimensions The distance between the points ( x 1 , y 1 , z 1 ) and ( x 2 , y 2 , z 2 ) is d x x y y z z ( ) ( ) ( ) . 2 1 2 2 1 2 2 1 2 = − + − + − • Midpoint Formula in Three Dimensions The midpoint of the segment with endpoints ( x 1 , y 1 , z 1 ) and ( x 2 , y 2 , z 2 ) is + + + M x x y y z z 2 , 2 , 2 1 2 1 2 1 2 .
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Example 1 Using Euler’s Formula
Euler’s Formula is verified using the number of vertices, edges, and faces of a polyhedron in this example. The pyramid has five vertices, eight edges, and five faces. Substitute these values into Euler’s Formula. The two sides of the formula are equal using the values obtained from the polyhedron, so the formula is verified for this object.
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