# 3D Polyhedra Viewer - Platonic & Archimedean Solids, Nets

> Rotate the 5 Platonic and 13 Archimedean solids, prisms and pyramids in 3D: truncation, duals, Euler’s formula, area and volume, unfolding and printable nets.

- URL: https://qilingtools.com/en/math/polyhedra
- Category: Math Tools (https://qilingtools.com/en/math)
- Tags: [Math](https://qilingtools.com/en/tags/math), [Education](https://qilingtools.com/en/tags/education), [Design](https://qilingtools.com/en/tags/design)
- Last updated: 2026-10-26
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

A hands-on 3D model kit for solid geometry. It includes the 5 Platonic solids (tetrahedron, cube, octahedron, dodecahedron, icosahedron), all 13 Archimedean solids (from the truncated tetrahedron and cuboctahedron to the football-shaped truncated icosahedron and the chiral snub cube and snub dodecahedron), and prisms, antiprisms, pyramids and bipyramids with any number of base sides. Drag to rotate, scroll to zoom and switch between solid, wireframe and transparent views, with faces coloured by their number of sides. Live counts of vertices V, edges E and faces F check Euler’s formula V − E + F = 2, alongside the face make-up (say, 8 triangles + 6 squares) and the vertex configuration such as 3.4.3.4. A truncation slider morphs a cube smoothly into a truncated cube and then a cuboctahedron, with one-click uniform truncation and rectification; the dual switch reciprocates about the midsphere, turning the cube into an octahedron, the dodecahedron into an icosahedron and Archimedean solids into Catalan solids. Enter an edge length a for surface area, volume and circumradius, with exact formulas for the Platonic solids. The unfold feature finds a non-overlapping net automatically and animates the solid opening out flat; the flat net takes an edge length in millimetres, adds glue tabs and downloads as SVG or an A4 PDF to print, cut and fold. Everything is computed in your browser.

## How to use

1. Pick a polyhedron from the list; for prisms and pyramids set the number of base sides n
2. Drag to rotate, choose solid, wireframe or transparent, and try the truncation slider or the dual switch
3. Read V, E, F and Euler’s formula, and enter an edge length for area and volume
4. Play the unfolding animation, or open the flat net, set the printed edge and download SVG or PDF

## Features

- 5 Platonic + 13 Archimedean solids, plus prisms, antiprisms, pyramids and bipyramids
- Drag to rotate; solid, wireframe and transparent views
- V, E, F counts with Euler’s formula check, face types and vertex configuration
- Continuous truncation (truncate → rectify) and dual polyhedra
- Surface area, volume and circumradius from the edge length, exact formulas for Platonic solids
- Automatic non-overlapping nets with an unfolding animation
- Printable nets with glue tabs and millimetre edge lengths, as SVG or A4 PDF

## FAQ

### Why are there only five Platonic solids?

At each vertex at least three identical regular polygons must meet, and their angles must add up to less than 360° for the corner to fold up. Equilateral triangles (60°) can meet in threes, fours or fives — tetrahedron, octahedron, icosahedron; squares (90°) only in threes — the cube; regular pentagons (108°) only in threes — the dodecahedron. Three hexagons already make exactly 360° and lie flat, so there are no more.

### Does Euler’s formula V − E + F = 2 hold for every polyhedron?

It holds for every convex polyhedron, and for any polyhedron without holes (one that can be inflated into a sphere). A cube gives 8 − 12 + 6 = 2 and a football (truncated icosahedron) 60 − 90 + 32 = 2. The tool checks it live for every solid, truncation and dual. A polyhedron with a hole through it, like a picture frame, gives 0 instead.

### What is the difference between truncation and rectification?

Truncation slices a little off every corner along the edges, so each old vertex becomes a new face; cut a cube until the new and old edges are equal and you get the truncated cube (3.8.8). Keep cutting to the midpoints of the edges and neighbouring cuts meet, the old edges vanish, and you have the rectification — for a cube, the cuboctahedron (3.4.3.4). The slider from 0 to 0.5 shows the whole journey.

### How do I build a paper model from the net?

Set the edge length in millimetres, download the PDF and print it on A4 at 100% (not “fit to page”). Cut along the solid lines, crease along the dashed lines, put glue on the grey tabs and stick each tab behind its matching edge. Large Archimedean nets may not fit on A4, so the PDF is scaled down and the tool tells you by how much; if a tab overlaps another part, just trim it narrower.

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