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, as if you unwrapped and flattened it). Pick a shape, adjust its dimensions, and both figures update from the exact formula for that shape — nothing here is approximated except where a shape (like the ellipse) genuinely has no simple closed-form perimeter.
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.
Three side lengths pin down a triangle's shape completely and let this tool compute an exact perimeter alongside the area (via Heron's formula) — base and height alone would give area but leave the third side, and therefore the perimeter, undetermined.
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.