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How do you determine the length of the space diagonal in a rectangular prism?

To calculate the length of the space diagonal in a rectangular prism, you can use the formula:

d=l2+w2+h2d = \sqrt{l^2 + w^2 + h^2}

where:

  • dd is the length of the space diagonal,
  • ll is the length of the prism,
  • ww is the width, and
  • hh is the height.

A rectangular prism is a three-dimensional geometric figure composed of six rectangular faces. The space diagonal is the segment that connects two opposite corners of the prism, traversing through its interior.

To find the length of this diagonal, you need to know the measurements of the prism’s three dimensions: length ll, width ww, and height hh.

The formula d=l2+w2+h2d = \sqrt{l^2 + w^2 + h^2} is derived from the application of the Pythagorean theorem in three dimensions. First, consider the base of the prism, which is a rectangle. The diagonal of this rectangle can be calculated using the Pythagorean theorem:

dbase=l2+w2d_{base} = \sqrt{l^2 + w^2}

This gives you the length of the diagonal across the base of the prism.

Next, visualize a right triangle formed by this base diagonal, the height hh of the prism, and the space diagonal dd. By applying the Pythagorean theorem once more to this right triangle, we can express the space diagonal dd as the hypotenuse:

d=dbase2+h2=(l2+w2)2+h2d = \sqrt{d_{base}^2 + h^2} = \sqrt{(\sqrt{l^2 + w^2})^2 + h^2}

Upon simplifying this expression, we arrive at:

d=l2+w2+h2d = \sqrt{l^2 + w^2 + h^2}

Thus, by squaring the lengths of the prism’s dimensions, summing these squares, and then taking the square root of that sum, you can determine the length of the space diagonal in a rectangular prism. This method provides an accurate measurement of the longest distance between any two points within the prism.

Answered by: Prof. Alan Smith
A-Level Physics Tutor
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