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.