Spatial Reasoning

A diagonal of a three-dimensional figure connects
two vertices of two different faces. Diagonal d
of a rectangular prism is shown in the diagram.
By the Pythagorean Theorem, l 2 + w 2 = x 2 , and
x 2 + h 2 = d 2 . Using substitution, l 2 + w 2 + h 2 = d 2 .
Diagonal of a Right Rectangular Prism
The length of a diagonal d of a right rectangular prism with length l,
width w, and height h is d = √
l 2 + w 2 + h 2 .
EXAMPLE 2 Using the Pythagorean Theorem in Three Dimensions
Find the unknown dimension in each figure.
A the length of the diagonal of a 3 in. by 4 in. by 5 in. rectangular prism
d = √ 3
2 + 4 2 + 5 2 Substitute 3 for l, 4 for w, and 5 for h.
= √
9 + 16 + 25
Simplify.
= √ 50 ≈ 7.1 in.
B the height of a rectangular prism with an 8 ft by 12 ft base and
an 18 ft diagonal
18 = √
8 2 + 12 2 + h 2 Substitute 18 for d, 8 for l, and 12 for w.
18 2 = ( √
8 2 + 12 2 + h 2 ) 2
324 = 64 + 144 + h 2
h 2 = 116
h = √ 116 ≈ 10.8 ft
Square both sides of the equation.
Simplify.
Solve for h 2 .
Take the square root of both sides.
2. Find the length of the diagonal of a cube with edge
length 5 cm.
Space is the set of all points in three dimensions.
Three coordinates are needed to locate a point in
space. A three-dimensional coordinate system has 3
perpendicular axes: the x-axis, the y-axis, and the
z-axis. An ordered triple (x, y, z) is used to locate a
point. To locate the point (3, 2, 4) , start at (0, 0, 0) .
From there move 3 units forward, 2 units right, and
then 4 units up.
EXAMPLE 3 Graphing Figures in Three Dimensions
Graph each figure.
A a cube with edge length 4 units and
one vertex at (0, 0, 0)
The cube has 8 vertices:
(0, 0, 0) , (0, 4, 0) , (0, 0, 4) , (4, 0, 0) ,
(4, 4, 0) , (4, 0, 4) , (0, 4, 4) , (4, 4, 4) .
10- 3 Formulas in Three Dimensions 671