1. What is a Parallelepiped?

A parallelepiped is a three-dimensional figure formed by six parallelograms. It's a prism whose faces are all parallelograms. When the radius from the center to each vertex is known, we can calculate its volume using trigonometric relationships.

2. How Does the Calculator Work?

The calculator uses the following formulas:

Where:

• \( V \) — Volume of the parallelepiped
• \( r \) — Radius from center to vertex
• \( \alpha, \beta, \gamma \) — Angles between radius vectors
• \( a, b, c \) — Edge lengths of the parallelepiped

Explanation: The formula calculates the edge lengths from the radius and angles, then computes the volume as the product of these lengths.

3. Importance of Radius in Calculation

Details: The radius defines the size of the parallelepiped relative to its central point. Combined with the angles between edges, it determines all dimensions of the shape.

4. Using the Calculator

Tips: Enter the radius and the three angles between edges (in degrees). All values must be positive (radius > 0, angles between 0° and 180°).

5. Frequently Asked Questions (FAQ)

Q1: What's the relationship between radius and edge lengths?
A: Each edge length is twice the radius multiplied by the sine of the corresponding angle between edges.

Q2: Can I use this for a rectangular box?
A: Yes, when all angles are 90°, it becomes a rectangular prism with edge lengths equal to 2r.

Q3: What units should I use?
A: Use consistent units - the result will be in cubic units of whatever radius unit you input.

Q4: How precise should my angle measurements be?
A: For accurate results, angles should be measured to at least one decimal place.

Q5: What if my angles don't add up properly?
A: The calculator assumes valid geometric angles. For a proper parallelepiped, certain angle relationships must be maintained.