Volume of Parallelepiped Calculator

Volume of Parallelepiped Calculator

Edge Lengths

Internal Angles (Degrees)

The total volume is:

400.00

cubic units

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Understanding the Volume of a Parallelepiped

A parallelepiped is a three-dimensional solid figure bounded by six faces, each of which is a parallelogram. You can think of it as the 3D version of a parallelogram, or a "squashed" cuboid. Calculating its volume depends on the lengths of its three intersecting edges and the angles between them.

The General Formula

For a general (oblique) parallelepiped with edge lengths a, b, and c and internal angles α (alpha), β (beta), and γ (gamma), the volume (V) is calculated using the following geometric formula:

V = abc × √[1 + 2cos(α)cos(β)cos(γ) – cos²(α) – cos²(β) – cos²(γ)]

Special Cases

• Rectangular Parallelepiped (Cuboid): When all angles are 90 degrees, the formula simplifies to the familiar V = a × b × c.
• Rhombohedron: When all edge lengths are equal (a = b = c).
• Cube: When all edges are equal and all angles are 90 degrees.

Example Calculation

Suppose you have a parallelepiped with the following dimensions:

• Edge a: 10 cm
• Edge b: 5 cm
• Edge c: 8 cm
• Angle α, β, γ: 90° (Rectangular)

Step 1: Multiply the lengths: 10 × 5 × 8 = 400.

Step 2: Since the angles are 90°, the square root term evaluates to 1.

Result: Volume = 400 cubic cm.

If the angles were 60° instead of 90°, the shape would be slanted (oblique), and the volume would be significantly less because the vertical height of the shape decreases as the angles deviate from 90 degrees.

Practical Applications

The volume of a parallelepiped is a fundamental concept in vector calculus (where it is calculated using the scalar triple product) and in mineralogy to describe crystal lattice structures. It is also used in structural engineering and physics to determine displacement and stress in skewed geometric components.

function calculateVolume() { var a = parseFloat(document.getElementById('edgeA').value); var b = parseFloat(document.getElementById('edgeB').value); var c = parseFloat(document.getElementById('edgeC').value); var alphaDeg = parseFloat(document.getElementById('angleAlpha').value); var betaDeg = parseFloat(document.getElementById('angleBeta').value); var gammaDeg = parseFloat(document.getElementById('angleGamma').value); if (isNaN(a) || isNaN(b) || isNaN(c) || isNaN(alphaDeg) || isNaN(betaDeg) || isNaN(gammaDeg)) { document.getElementById('volumeVal').innerHTML = "Invalid Input"; return; } // Convert degrees to radians var radAlpha = alphaDeg * (Math.PI / 180); var radBeta = betaDeg * (Math.PI / 180); var radGamma = gammaDeg * (Math.PI / 180); var cosA = Math.cos(radAlpha); var cosB = Math.cos(radBeta); var cosG = Math.cos(radGamma); // Formula for volume of general parallelepiped // V = abc * sqrt(1 + 2*cosA*cosB*cosG – cosA^2 – cosB^2 – cosG^2) var termInsideSqrt = 1 + (2 * cosA * cosB * cosG) – (cosA * cosA) – (cosB * cosB) – (cosG * cosG); if (termInsideSqrt < 0) { document.getElementById('volumeVal').innerHTML = "Invalid Angles"; return; } var volume = a * b * c * Math.sqrt(termInsideSqrt); document.getElementById('volumeVal').innerHTML = volume.toLocaleString(undefined, { minimumFractionDigits: 2, maximumFractionDigits: 2 }); } // Initialize calculation window.onload = function() { calculateVolume(); };