Cross Section Vizualizer for Solids

Math Geometry • Three Dimensional Geometry

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

3D Solid Cross-Section Calculator & Visualizer

Visualize a plane cutting a 3D solid and compute the cross-section area. Slices of a sphere are computed with an exact formula; other solids use a fast, accurate sampling estimate that you can refine with grid resolution.

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

Solid

Type: Center \(c_x\): Center \(c_y\): Center \(c_z\):

Sphere radius \(R\):

Cutting plane

Plane mode:

Plane height \(k\):

This is the plane \(z=k\) in world coordinates.

Cut slider (moves the plane):

Range is auto-scaled to the solid size.

Accuracy & view

Sampling grid (for non-sphere / oblique cuts): Projection: Show axes: Show ground grid: Show plane fill: Display precision:

Drag to orbit • Shift+drag to pan • wheel/trackpad to zoom. “Reset view” fits the solid. Units are square.

Ready

3D view (interactive • square units)

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Theory

What a cross-section is

A cross-section of a 3D solid is the 2D region you get when you “slice” the solid with a plane. If you imagine a CT scan or a saw cut through a material, each slice is a cross-section. Mathematically, if a solid is a set \(S \subset \mathbb{R}^3\) and the cutting plane is a set \(P\), then the cross-section is the intersection \(S \cap P\). The quantity we typically want is the area of that planar region.

A plane can be described in many equivalent ways. A common form is a normal vector \(\hat n\) (a unit vector perpendicular to the plane) plus a point on the plane. In this calculator, the plane is positioned relative to the solid’s center \(C\) using \[ \hat n \cdot (x - C) = d, \] where \(d\) is a signed offset (positive means the plane is shifted in the \(+\hat n\) direction). The special horizontal mode \(z=k\) is just the case \(\hat n=(0,0,1)\) and \(d=k-c_z\).

Sphere slice formula (exact)

The nicest cross-section to compute is the sphere. Consider a sphere of radius \(R\) centered at \(C\). Any plane that cuts the sphere produces a circle (or a single point if tangent, or nothing if the plane misses the sphere). Let \(d\) be the signed distance from the center to the plane (in the calculator, this is exactly the offset in the normal mode). The radius of the circular cross-section is found from a right triangle:

\[ \begin{aligned} r_{\text{sec}}^2 + d^2 &= R^2 \\ r_{\text{sec}} &= \sqrt{R^2 - d^2}. \end{aligned} \]

Therefore, the cross-section area is \[ \begin{aligned} A &= \pi r_{\text{sec}}^2 \\ &= \pi (R^2 - d^2). \end{aligned} \] This result is exact, works for any plane orientation, and immediately tells you when there is no intersection: if \(|d|>R\), then \(R^2-d^2<0\) and the plane does not meet the sphere.

Other solids and why sampling is used

For solids like cylinders, cones, and cuboids, a plane cut can produce shapes that change depending on the orientation: circles, ellipses, rectangles, trapezoids, or more general polygons. Some special cases have simple formulas (for example, a cylinder cut by a plane perpendicular to its axis gives a circle of area \(\pi r^2\)), but a fully general “any plane, any solid” closed-form solution is not always convenient to implement in a single interactive tool.

This calculator therefore uses a fast sampling method for non-sphere cases (and you can increase the grid for higher accuracy). The idea is to build a 2D coordinate system \((s,t)\) on the cutting plane using two orthonormal directions \(u\) and \(v\) that lie in the plane. Then points on the plane are represented as \[ x(s,t)=P_0 + su + tv, \] where \(P_0\) is a point on the plane. The tool samples many \((s,t)\) points on a square window, checks whether each corresponding 3D point lies inside the solid, and estimates the area by counting how many samples are inside. With a sufficiently fine grid, this produces an accurate approximation and also provides a boundary outline for visualization.

In practice, you should increase the grid resolution when the cross-section boundary is highly curved (oblique cuts on cones/cylinders) or when the cross-section is small compared to the solid. The 3D visualization is meant to help you verify that the plane is positioned where you expect and that the intersection curve looks reasonable.

Frequently Asked Questions

What is the cross-section area of a 3D solid?

It is the area of the 2D region formed where a plane intersects a 3D solid. The cross-section depends on both the solid shape and the plane position and orientation.

How do you find the area of a sphere slice by a plane?

If the sphere has radius R and the plane is at signed distance d from the center, the cross-section is a circle with radius r = sqrt(R^2 - d^2). The area is A = pi (R^2 - d^2), and there is no intersection if |d| > R.

Why is the cylinder, cone, or cuboid cross-section area approximate?

For general plane orientations the cross-section shape can be complex, so the tool estimates area by sampling points on the plane and counting those that fall inside the solid. Increasing the grid resolution improves accuracy.

What plane equation does the normal-and-offset mode use?

The tool uses n_hat dot (x - C) = offset, where C is the solid center and n_hat is the unit normal. The slider changes the offset, moving the plane along the normal direction.

3D Solid Cross-Section Calculator & Visualizer

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