Perimeter is the distance around a 2D shape; surface area is the skin of a 3D solid. Different questions despite the similar names.
Scale every dimension by k: area by k², volume by k³. Doubling a shape does not just double the material.
Flattenable without stretching or tearing: cylinders, cones and every flat-faced solid. Spheres and tori are not, which is why every flat map of Earth distorts something.
Any solid without holes. Counted live from the mesh actually drawn, so it is a check rather than a claim.
Cones and pyramids both carry a ⅓ against the matching cylinder or prism. It is one fact: three copies of the tapering solid fill the straight one exactly, whatever the base shape. A prism has the same cross-section all the way through (volume = base area × length); a pyramid tapers to a point.
a² + b² = c² says the square on the hypotenuse equals the two squares on the legs together. It only holds with a right angle between a and b; the cosine rule is Pythagoras with the correction −2ab·cos C bolted on.
Straight line 180°, round a point 360°, triangle 180°, quadrilateral 360°, n-gon (n−2)×180°, exterior angles always 360°. Parallel lines: Z equal, F equal, C adds to 180°. Circles: the centre sees double, a diameter sees a right angle.
The 3D view defaults to orthographic: parallel edges stay parallel and a cube’s far face is drawn as large as its near one, which technical drawing uses because it compares faces honestly. Perspective looks like a photograph and reads depth better on rounder solids, but makes the near side larger: a switch, not the default.
An ellipse’s perimeter has no elementary closed form: it needs an elliptic integral. The tool uses Ramanujan’s second approximation, accurate to a fraction of a percent for any realistic ellipse, and the ellipsoid’s surface area uses the Knud Thomsen approximation; both are labelled as approximations wherever they appear.
A real hinge fold: a spanning tree over the faces, each rotating about the actual edge it shares with its parent by exactly the angle that makes them coplanar, so the flat net’s area matches the surface-area formula; you can stop it half way and rotate. A surface can be flattened only if it has zero Gaussian curvature everywhere (cones, cylinders, prisms, pyramids do, despite the cone looking curved); a sphere curves the same way in every direction at once. Not a limitation of the tool.
Unlike a circle, an ellipse's perimeter genuinely has no elementary closed-form expression — it requires an elliptic integral. This tool uses Ramanujan's second approximation, which is accurate to within a fraction of a percent for any realistic ellipse.
A prism has the same cross-section all the way through (like a Toblerone bar) — its volume is simply the base area times the length. A pyramid tapers to a single point, which is exactly why its volume formula has that extra ⅓ factor.
Area is fundamentally a product of two lengths (like base × height); volume is a product of three. Scale every dimension of a shape by a factor k, and its area scales by k², its volume by k³ — which is why doubling a shape's size doesn't just double how much material or space it needs.
It is a real hinge fold, not a visual effect. The tool builds a spanning tree over the solid's faces, and every face rotates about the actual edge it shares with its parent face — by exactly the angle that makes the two faces coplanar. At the end of the animation the whole surface lies flat in one plane, and the total area of that net matches the surface-area formula. Because the fold is genuine, you can stop it half way and rotate the half-open shape.
It's not a limitation of this tool — it's a real fact about those shapes. A surface can only be flattened without stretching or tearing if it has zero Gaussian curvature everywhere, which cones, cylinders, prisms and pyramids all do (despite the cone looking curved, it's flat in the direction that matters). A sphere curves the same way in every direction at once, the same reason every flat map of Earth distorts something — area, shape, or distance — because the globe itself cannot be flattened.
Releases in which this page changed, newest last. Derived from the archived copy of every release, not from notes written afterwards — so it reflects what actually shipped. Site-wide passes are left out; they are in the full changelog.