Space Diagonal of Cuboid given Total Surface Area, Width, and Height Calculator

Formula Used:

\[ d = \sqrt{\frac{TSA}{2} - (w \cdot h) + w^2 + h^2} \]

Space Diagonal (d):

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1. What is Space Diagonal of Cuboid?

The space diagonal of a cuboid is the longest straight line that can be drawn inside the cuboid, connecting two opposite vertices. It passes through the interior of the cuboid and represents the maximum distance between any two points within the three-dimensional shape.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ d = \sqrt{\frac{TSA}{2} - (w \cdot h) + w^2 + h^2} \]

Where:

\( d \) — Space diagonal of the cuboid
\( TSA \) — Total surface area of the cuboid
\( w \) — Width of the cuboid
\( h \) — Height of the cuboid

Explanation: This formula derives the space diagonal from the total surface area and two of the three dimensions (width and height) of the cuboid.

3. Importance of Space Diagonal Calculation

Details: Calculating the space diagonal is important in various fields including architecture, engineering, and packaging design. It helps determine the longest internal measurement of a rectangular prism, which is crucial for fitting objects through openings, designing storage spaces, and understanding spatial relationships.

4. Using the Calculator

Tips: Enter the total surface area in square units, width and height in linear units. All values must be positive numbers. The calculator will compute the space diagonal of the cuboid.

5. Frequently Asked Questions (FAQ)

Q1: What is the relationship between space diagonal and other cuboid dimensions?
A: The space diagonal is related to all three dimensions through the formula: d² = l² + w² + h², where l is length, w is width, and h is height.

Q2: Can this formula be used for any rectangular prism?
A: Yes, this formula applies to all cuboids (rectangular prisms) regardless of their specific dimensions.

Q3: How accurate is this calculation?
A: The calculation is mathematically exact when the input values are precise and the cuboid dimensions satisfy the surface area formula.

Q4: What units should I use for the inputs?
A: Use consistent units throughout - square units for surface area and the same linear units for all dimensions.

Q5: Can I calculate the length if I know the space diagonal and other dimensions?
A: Yes, you can rearrange the formula to solve for any missing dimension when the space diagonal and other dimensions are known.