Prism Calculator

Math Geometry • Three Dimensional Geometry

Written by STEM Calculators Team Published February 1, 2026 Updated February 24, 2026

Prism Volume Calculator – Right, Oblique, Cube & Cuboid (3D Shapes)

Compute volume \(V=A_b\,h_\perp\) and surface area for right and oblique prisms. Includes special cases: cube (\(V=a^3\)) and cuboid (\(V=lwh\)). For oblique prisms, the top face is a translation of the base, so lateral faces are parallelograms.

Inputs accept 1e-3, pi, e, sqrt(2), sin(), cos(), tan(), ln(), log(), abs(). Use * for multiplication.

Base shape

Base type:

Length \(l\): Width \(w\):

Base area \(A_b\) and perimeter \(P\) are computed from the chosen base. These drive volume and surface area.

Height & prism type

Prism type: Perpendicular height \(h_\perp\): Display precision: Units label (optional):

3D view

Projection: Show axes: Fill faces lightly:

Drag to orbit • Shift+drag to pan • wheel/trackpad to zoom. Double-click to fit.

Ready

3D prism view (interactive)

Highlighted base edge shows a key base dimension (e.g., \(l\) or \(a\)). The vertical segment shows \(h_\perp\). A slanted lateral edge shows \(h_s\) in oblique mode.

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Theory

What a prism is (right vs oblique)

A prism is a 3D solid formed by taking a 2D base polygon and creating a second, congruent polygon parallel to it (the “top”), then connecting corresponding vertices. The base and top are parallel faces, and the other faces are called the lateral faces. In most practical models, the top is obtained by translating the base by a single constant vector. That translation viewpoint is very useful because it guarantees every lateral edge is parallel and every lateral face is a parallelogram.

A right prism has lateral edges perpendicular to the base plane. Visually, it looks like a straight extrusion of the base upward. An oblique prism is “skewed”: lateral edges are tilted and not perpendicular to the base plane, but the top face is still parallel to the base. Many real objects (skewed containers, angled supports, or a box pushed sideways as you lift it) behave like oblique prisms.

Volume: \(V = A_b \, h_\perp\) (why it stays the same when skewed)

The key prism formula is \[ V = A_b \, h_\perp, \] where \(A_b\) is the base area and \(h_\perp\) is the perpendicular height (the distance between the parallel base and top planes). This remains true for both right and oblique prisms. The reason is that the volume depends on the area of horizontal cross-sections and how far those sections extend in the perpendicular direction. If the top face is a translation of the base, then every cross-section parallel to the base has the same area \(A_b\), so the solid is essentially a “stack” of identical slices whose thickness totals \(h_\perp\).

This explains an important practical fact: skewing a prism changes its surface appearance but not its capacity if the base area and perpendicular height are unchanged. That’s why a slanted box can still hold the same volume as a straight one.

Surface area: right prisms vs oblique prisms

The total surface area of any prism is \[ A_T = 2A_b + A_L, \] where \(A_L\) is the lateral surface area (the sum of all lateral face areas). For a right prism, each lateral face is a rectangle with area (edge length)\(\times h_\perp\). Summing over all edges gives the shortcut \[ A_L = P\,h_\perp, \] where \(P\) is the perimeter of the base.

For an oblique prism, lateral faces are parallelograms, and their areas can depend on the direction of the skew. If the top is translated by vector \(\mathbf{L}=(s_x,s_y,h_\perp)\), and a particular base edge vector is \(\mathbf{e}_i\), then the corresponding parallelogram face area is \[ A_i = \lVert \mathbf{e}_i \times \mathbf{L} \rVert. \] This tool uses that exact vector formula so the lateral area responds correctly when you change the offset direction \(\varphi\). In special cases (for example, when the skew is aligned so it doesn’t “shear” a face), the right-prism shortcut may match closely, but in general \(A_L\neq P\,h_\perp\) for oblique prisms.

Special cases: cuboid and cube

A cuboid is a rectangular right prism with dimensions \(l\), \(w\), and \(h\). Its base area is \(A_b=lw\), so \[ V = lwh, \] and its total surface area is \(A_T=2(lw+lh+wh)\). A cube is the special case \(l=w=h=a\), giving \(V=a^3\) and \(A_T=6a^2\). These are so common that the calculator includes quick presets, but they are still just prisms: the same “base area times perpendicular height” logic applies.

When using units, remember: length uses units (e.g., m), area uses squared units (m²), and volume uses cubic units (m³). This calculator keeps those distinctions clear in the output.

Frequently Asked Questions

Does an oblique prism have the same volume as a right prism?

Yes, if the base area is the same and the perpendicular height (distance between the parallel base and top planes) is the same. Volume uses V = Ab x h_perp, not the slant edge length.

What is the difference between perpendicular height and slant height in a prism?

Perpendicular height h_perp is the shortest distance between the base and top planes. Slant height (or slant edge length) is the length of a tilted lateral edge and is larger than h_perp when the prism is oblique.

How do you compute the base area of a regular n-gon from side length?

For a regular n-gon with side length a, the area is Ab = (n a^2) / (4 tan(pi/n)). The perimeter is P = n a.

How do I enter a custom polygon base in the calculator?

Choose the custom polygon option and enter vertices as (x,y) or x,y, one point per line, in order around the shape. The calculator uses the shoelace method for area and sums edge lengths for perimeter.

Why is lateral surface area different for an oblique prism?

In a right prism each lateral face area equals (base edge length) x h_perp, so AL = P x h_perp. In an oblique prism, lateral faces are parallelograms whose areas depend on the slant direction, so the total lateral area generally changes with the offset.

Prism Volume Calculator – Right, Oblique, Cube & Cuboid (3D Shapes)

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Frequently Asked Questions

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