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Question

An illustration of a rectangular prism is shown with labeled parts. The top, front and right sides are visible. The bottom, left, and rear sides are not visible, but their positions are indicated by dashed lines. The vertical edge joining the front and left sides shows a height of 12 centimeters. The front base edge shows a width of 3 centimeters. The right base edge shows a length of 4 centimeters. A dotted line extends diagonally from the vertex at the bottom back right side of the prism through the interior of the prism to the vertex at the top front left.

Find the length of the diagonal of the prism.

Answer 1

To find the length of the diagonal of the rectangular prism, we use the formula for the diagonal \(d\) of a rectangular prism, given by:

\[ d = \sqrt{l^2 + w^2 + h^2} \]

where:

• \(l\) is the length,
• \(w\) is the width, and
• \(h\) is the height of the prism.

From the problem statement, we have:

• The height \(h = 12\) cm,
• The width \(w = 3\) cm,
• The length \(l = 4\) cm.

Now, we can plug these values into the formula for the diagonal:

\[ d = \sqrt{(4)^2 + (3)^2 + (12)^2} \]

Calculating each term inside the square root:

\[ d = \sqrt{16 + 9 + 144} \]

Now adding these values together:

\[ d = \sqrt{169} \]

Finally, taking the square root:

\[ d = 13 , \text{cm} \]

Therefore, the length of the diagonal of the prism is \( \boxed{13} \) centimeters.