Volume Of A Parallelepiped Calculator - Edges and Angles

The volume of a parallelepiped calculator returns the 3D capacity of a slanted box from three edge lengths and three angles, then displays the answer in m^3, cm^3, ft^3, in^3, liters, or US gallons. It is the most general box-shaped solid: 90 deg on all three angles reduces the answer to L*W*H, and three equal edges with three 60 deg angles returns a rhombohedron.

• • Rhombic unit cell in crystallography: Compute the volume of a triclinic or rhombohedral crystal unit cell from its three lattice constants and three angles, then divide molar mass by that volume.
• • Sheared storage tank or hopper: Estimate the capacity of a tank whose walls lean inward or outward, where the inside space is a parallelepiped rather than a cuboid, and the angle drives how much liquid fits.
• • Cross-product homework or exam problem: Verify a vector-based answer to V = |a . (b x c)| by feeding the same edges and angles into the three-edge formula and comparing the numeric result.
• • Lofted 3D model in CAD or animation: Compute the volume of an oriented bounding box around a non-axis-aligned mesh used in 3D rendering, collision detection, or shipping-package design.

A parallelepiped is the most general parallelogram-faced solid. Each of its six faces is a parallelogram, its twelve edges group into three parallel families of four, and the only special case most people meet in school is a cuboid, which is just a parallelepiped with three right angles.

The calculator converts the three edges to meters, converts the three angles from degrees to radians, and combines them with a single trigonometric factor. When that factor is 1, the parallelepiped is a right-angled box and the result is exactly a*b*c; when it is below 1, the box is sheared.

• a: First edge length of the parallelepiped, in the chosen length unit.
• b: Second edge length, in the same unit as a, perpendicular to a only in the orthogonal case.
• c: Third edge length, in the same unit as a, perpendicular to both a and b only in the orthogonal case.
• alpha: Angle in degrees between edges b and c, defaulting to 90 deg for the cuboid shortcut.
• beta: Angle in degrees between edges a and c, defaulting to 90 deg for the cuboid shortcut.
• gamma: Angle in degrees between edges a and b, defaulting to 90 deg for the cuboid shortcut.

This trig factor is the Cayley-Menger-style correction for a parallelepiped: it equals 1 for an orthogonal box, sqrt(0.5) ~ 0.7071 for a rhombohedron with three 60 deg angles, and it approaches 0 as any one angle heads to 180 deg.

The result is computed in cubic meters and re-scaled to the chosen display unit with exact factors: 1 m^3 = 1,000,000 cm^3 = 35.3146667215 ft^3 = 61,023.7440947 in^3 = 1,000 L = 264.1720523581 US gal.

According to Wolfram MathWorld, the volume of a parallelepiped with edge lengths a, b, c and included angles alpha, beta, gamma is a*b*c * sqrt(1 - cos^2 alpha - cos^2 beta - cos^2 gamma + 2*cos(alpha)*cos(beta)*cos(gamma)), the exact expression the calculator evaluates on every input change.

When all three angles are 90 degrees the parallelepiped collapses to a box-shaped solid and the result matches the cuboid volume calculator.

Four small ideas explain every result on this page and stop the most common mix-ups when working with slanted, skewed, or sheared boxes.

These four ideas explain the result panel layout. The product a*b*c is the bounding-box volume, the trig factor is the share the parallelepiped fills, and their product is the answer.

A rectangular prism is the same box-shaped solid and the rectangular prism volume calculator confirms that V = L*W*H is just the alpha=beta=gamma=90 deg row of the parallelepiped formula.

Follow five short steps, switching the length or volume unit at any time without re-entering dimensions.

1. 1 Enter edges a, b, and c: Type the three edge lengths in the chosen length unit. The three values must use the same unit, and any of them can be zero for a flat case.
2. 2 Enter angles alpha, beta, and gamma: Type the three angles between the edges in degrees. Use 90 deg on all three to switch the tool into the cuboid shortcut.
3. 3 Pick the length unit: Select mm, cm, m, in, ft, or yd to match the numbers you typed. The page converts to meters internally.
4. 4 Pick the volume unit: Switch the volume output between m^3, cm^3, ft^3, in^3, liters, and US gallons, and confirm the unit label in the result panel.
5. 5 Read the volume and trig factor: Watch the volume, the canonical a*b*c in m^3, and the trig factor update in real time as any input changes.

For a 60 cm x 40 cm x 30 cm storage bin, set the length unit to centimeters, type 60, 40, 30, leave the three angles at 90 deg, and pick liters. The result panel returns 72 L of capacity and a trig factor of 1.

For a vector-based parameterization, the cross product calculator returns the perpendicular vector b x c whose magnitude feeds the same |a . (b x c)| expression.

A dedicated volume of a parallelepiped calculator removes the algebra from one closed-form expression and keeps unit handling consistent across metric and imperial work.

• • Handles any skew, shear, or tilt in one place: Type the three edges and three angles and the result panel shows the volume, the bounding-box product, and the trig factor together, so a cuboid, a rhombohedron, and a triclinic cell live in the same form.
• • Six volume units without re-entering values: Switch the volume output between m^3, cm^3, ft^3, in^3, liters, and US gallons without re-entering dimensions, avoiding the most common unit-conversion error between lab, classroom, and shipping numbers.
• • Cuboid shortcut is the default: The default angles are 90 deg, so a 2 m x 3 m x 4 m entry returns the familiar 24 m^3 and the trig factor reads 1. Type any non-right angle and the result drops below a*b*c as the formula predicts.
• • Canonical a*b*c in m^3 alongside the answer: The edge product row stays in cubic meters, making it easy to spot when a unit switch has gone wrong, when a rhombohedron gives the expected a^3/sqrt(2) drop, or when the solid flattens toward zero volume.
• • Hand-off to box, cube, and 3D peers: Set alpha=beta=gamma=90 deg and the result matches the cuboid volume calculator. With three equal edges and three equal angles, the cube volume calculator returns a^3 as the rhombohedron collapses to a cube.

If a=b=c with all 90 deg angles, the parallelepiped reduces to a cube of side a and the cube volume calculator returns a^3 with one input.

Three measurable factors control the precision of every result, and a few practical limits apply to real-world slanted boxes.

• • The calculator assumes the angles are the three included angles between the edges, not the face angles. CAD packages sometimes report face angles, which are the supplements of the included angles and need a 180 deg subtraction before typing them in.
• • The closed-form expression assumes a perfect parallelepiped with three pairs of parallel faces. Real objects have rounded edges, bevels, or material thickness, so the true inside volume is usually below the a*b*c * trig factor result.

As published by Wikipedia, the volume of a parallelepiped equals the absolute value of the scalar triple product |a . (b x c)| of its three edge vectors, the same value the three-edge, three-angle formula returns when only edge lengths and the angles between them are given.

According to Encyclopaedia Britannica, a parallelepiped is a three-dimensional figure whose six faces are all parallelograms, and a rectangular parallelepiped is the special case where those six faces are rectangles, the everyday cuboid used in storage and shipping.

An ellipsoid is the smooth cousin of an oblique parallelepiped and the ellipsoid volume calculator returns the matching 4/3*pi*a*b*c value for an axis-aligned solid.