Find the length of the diagonal of a rectangular prism of sides three centimeters, four centimeters, and six centimeters. Round your answer to the nearest hundredth.
The best way to begin a question like this is by sketching a diagram. So here we have a rectangular prism, or cuboid, with sides of three centimeters, four centimeters, and six centimeters. It really doesn’t matter which way round we draw this rectangular prism or which sides are three, four, or six centimeters because the diagonals will all be the same length. An example of one of the diagonals that we’re looking for is the space diagonal, which runs from vertex 𝐺 to vertex 𝐴. An example of a face diagonal would be this line segment in pink running from vertex 𝐶 to vertex 𝐴. This type of diagonal is a two-dimensional one and not the one that we need to calculate in this question.
We can find the length of any of the diagonals by applying the three-dimensional extension of the Pythagorean theorem. This extension states that for a cuboid, or rectangular prism, with side lengths 𝑎, 𝑏, and 𝑐 and a diagonal of length 𝑑, then 𝑎 squared plus 𝑏 squared plus 𝑐 squared equals 𝑑 squared. We can define any of the side lengths with the letters 𝑎, 𝑏, and 𝑐. So let’s take 𝑎 is equal to three centimeters, 𝑏 is equal to four centimeters, and 𝑐 is equal to six centimeters. We are going to calculate the unknown length of the diagonal 𝑑.
Substituting these into the three-dimensional extension, we have three squared plus four squared plus six squared equals 𝑑 squared. On the left-hand side, we can square the values to give nine plus 16 plus 36, which is 61. And of course, we don’t want to find 𝑑 squared; we want to find 𝑑. So we take the square root of both sides. Because 𝑑 is a length, we can ignore the negative solution of this.
Sometimes we might need to leave our answer in this square root form or a simplified version of it. But here, since we’re asked for the answer to the nearest hundredth, we’ll need to find a decimal equivalent. The square root of 61 is 7.810 and so on. So rounding the answer to the nearest hundredth and including the units, we get the answer of 7.81 centimeters.
We can see how we could have found the length of any of these space diagonals and got the same value. These diagonals are all contained in the same rectangular prism with side lengths three, four, and six. And therefore, the value of 𝑑 will always be the same.