10.1.3 Formulas in Three Dimensions (continued)
Example 2 Using the Pythagorean Theorem in Three Dimensions
The length of the diagonal of the rectangular prism can be found using the formula given here. Alternatively, the length of the diagonal of the rectangular prism can be found by using the Pythagorean Theorem twice. First, the Pythagorean Theorem will be used to find the diagonal of the rectangular prism’s base (the diagonal that passes through the base of the prism). Notice that the diagonal of the rectangular prism’s base is the hypotenuse of a right triangle where the legs are the length and width of the rectangular prism’s base (7 in. and 10 in.). Use the Pythagorean Theorem to find the length of that hypotenuse. c 2 = 7 2 + 10 2 , c 149 = in. So, the length of the diagonal of the rectangular prism’s base is 149 in. Now use that length to find the length of the rectangular prism’s diagonal (the diagonal that passes through the interior of the prism). Notice that the rectangular prism’s diagonal is the hypotenuse of a right triangle where the legs are the base’s diagonal and the length of the rectangular prism (5 in.). So, use the Pythagorean Theorem to find the length of the prism’s diagonal. d d 149 5 , 174 in. 2 2 2 = + = The height of a rectangular prism is determined using the Pythagorean Theorem. The length (5 ft) and width (3 ft) of the bases and the length of the diagonal (12 ft) are given. To find the height of the rectangular prism, substitute the known lengths into the formula for the length of the diagonal and solve for the unknown height. The height of the prism is the square root of 110 feet.
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