Every shape here reduces to the same handful of ideas: a 2D shape has an area (how much surface it covers) and a perimeter (the distance around its edge); a 3D solid has a volume (how much space it fills) and a surface area (the total area of its outer skin). Pick a shape, adjust its dimensions, and both figures update from the exact formula — nothing here is approximated except where a shape genuinely has no elementary closed form, which is flagged where it happens. Every solid can be rotated by dragging, and every solid that can be flattened will fold flat along its real hinges while you watch.
A sphere and a cube with the same volume don't have the same surface area — the sphere always has less. A sphere of radius 3 has a volume of about 113 cubic units; a cube with that same volume needs sides of about 4.84 units, giving it a surface area near 140.5 — noticeably more than the sphere's 113. That's not a coincidence: for a fixed volume, a sphere always has the least possible surface area of any shape.
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.