Calculate the cross-sectional area, shape, and perimeter when slicing 3D solids (sphere, cylinder, cone, cube, rectangular prism, ellipsoid) at any height or angle.
A cross-section is the 2D shape you see when you slice through a three-dimensional solid with a flat cutting plane. Understanding cross-sections is fundamental in mathematics, physics, engineering, and manufacturing — from MRI scans that slice through the human body to CNC machine paths cutting through metal stock.
The shape and size of a cross-section depend on two things: the geometry of the original solid and the orientation of the cut. A horizontal slice through a sphere always produces a circle, but the radius of that circle depends on how far the cutting plane is from the center. A vertical cut through a cylinder produces a rectangle, while a horizontal cut produces a circle. Even a simple cube can yield squares, rectangles, triangles, or hexagons depending on how you orient the cut.
This calculator supports six common 3D solids: sphere, cylinder, cone, cube, rectangular prism, and ellipsoid. For each, you specify the solid's dimensions and the position of the cutting plane. The tool then computes the cross-section's shape, area, perimeter, and dimensions. It also shows the solid's total volume and the ratio of your cross-section area to the maximum possible cross-section. Presets based on real objects (soccer balls, soup cans, dice, shoeboxes) let you explore quickly. Whether you're studying for a geometry exam, designing a 3D-printed part, or estimating material cross-sections for structural analysis, this calculator gives you instant, accurate results.
This cross calculator reduces manual rework when you need quick checks for assignments, exam prep, and design calculations. You can enter 3D Solid, Cutting Plane, Unit and immediately see dependent measurements, validity checks, and geometry relationships in one place. That makes it easier to catch input mistakes early and confirm your final answer before moving to the next step.
Sphere (distance d from center): CS radius = √(R² − d²), Area = π(R² − d²) Cylinder (horizontal): Area = πr², Perimeter = 2πr Cylinder (vertical): Area = 2r × h, Perimeter = 2(2r + h) Cone (height d from base): CS radius = r(1 − d/H), Area = πr²(1 − d/H)² Cube (parallel to face): Area = s² Rect. Prism (horiz.): Area = l × w Ellipsoid (distance d from center along c): semi-axes = a√(1−d²/c²), b√(1−d²/c²)
Result: For shape=sphere, r=11, h=5.5, the tool returns the solved cross outputs shown in the result cards.
This example uses a realistic input set from the calculator workflow. After entry, the calculator applies the built-in cross formulas and reports derived values, checks, and classifications automatically.
This page is tailored to cross, with outputs tied directly to the form fields (3D Solid, Cutting Plane, Unit). Instead of a one-line formula dump, it consolidates validation, derived metrics, and interpretation so you can solve and verify in one pass.
Use this tool for homework checks, worksheet generation, tutoring walkthroughs, and quick engineering geometry estimates. Presets and visual output blocks make it easier to compare scenarios and understand how each input affects the final result.
Keep units consistent, match each value to the correct field, and watch validity indicators before using the final numbers. If your case looks off, change one input at a time and use the output details to identify the mismatch quickly.
A cross-section is the 2D shape formed when a flat plane intersects a 3D solid. For example, slicing a sphere horizontally produces a circle.
Yes. A horizontal cut through a cylinder gives a circle, but a vertical cut gives a rectangle, and an angled cut gives an ellipse.
A cube can have square, rectangular, triangular, pentagonal, or hexagonal cross-sections depending on the angle and position of the cutting plane. Use this as a practical reminder before finalizing the result.
It determines a beam's strength (moment of inertia), fluid flow through pipes (flow rate = velocity × area), and heat transfer through walls. Understanding this concept helps you apply the calculator correctly and interpret the results with confidence.
A great circle — a cross-section through the center — with area πr². Any cut away from the center produces a smaller circle.
Yes. Enter the three full axis lengths (a, b, c) and the distance from center. The cross-section is an ellipse whose semi-axes scale by √(1 − d²/c²).