Areas and volumes of common shapes
disc: A = πr², circumference: P = 2πr
Eighteen shapes, from the most schoolbook to the less common: square, rectangle, triangle, circle, ellipse, trapezoid, parallelogram, rhombus, circular sector and regular polygon; cube, box, sphere, cylinder, cone, pyramid, torus and spherical cap. Every result comes with its formula and, for solids, the lateral area distinguished from the total area. When a quantity is not determined by the requested dimensions (the perimeter of a triangle known only by base and height) the tool says so instead of inventing a number.
226.195
- Formula
- V = πr²h, A = 2πr(r + h)
Useful reminder: doubling a length quadruples the area and octuples the volume. That is why an animal twice as tall weighs eight times more but has only four times the bone cross-section.
Scientific dossier
What the tool computes, what it assumes, where it stops being valid, and where its data comes from.
Method & formulasdisc: A = πr², circumference: P = 2πr
disc: A = πr², circumference: P = 2πr
sphere: V = 4πr³/3, A = 4πr²
cylinder: V = πr²h, cone: V = πr²h/3
pyramid: V = (base area) × h / 3
Two regularities are worth remembering rather than eighteen formulas. First, every “pointed” solid (cone, pyramid) is exactly one THIRD of the straight solid with the same base and height. One idea for two formulas. Second, the sphere’s surface is the derivative of its volume with respect to the radius: 4πr² is what you add by inflating the ball by an infinitesimal thickness. These links are not curiosities: they are what the tool checks in its tests, since a wrong formula would necessarily break them.
- Lateral area vs total area
- · the lateral area is that of the curved surface alone (the “wrap” of a cylinder); the total area adds the bases. The distinction matters: painting a pipe and painting a tin can do not need the same amount.
- Slant height of a cone
- · the distance from apex to base edge, l = √(r² + h²). Never the height. It is what enters the lateral area πrl, and confusing it with h is the most frequent cone mistake.
- Spherical cap
- · the portion of a ball cut off by a plane. Its curved surface is 2πRh regardless of where you cut, a result of Archimedes. Which makes two slices of equal thickness exactly equivalent in surface.
Validity domainUnits are the ones you enter, the tool converts nothing: everything must be in the same unit, and the result comes out in its square or cube.
Units are the ones you enter, the tool converts nothing: everything must be in the same unit, and the result comes out in its square or cube. The ellipse perimeter is the only NON-exact result here: it has no elementary closed form, and Ramanujan’s approximation used stays below 5·10⁻⁵ relative error (exact for a circle). Shapes are assumed perfect and regular; a dented real object, an irregular polygon or a truncated cone need other formulas.
Common pitfall: area and volume do not scale like lengthDoubling a dimension doubles neither the area nor the volume: area is multiplied by 4, volume by 8.
Doubling a dimension doubles neither the area nor the volume: area is multiplied by 4, volume by 8. That is why a 40 cm pizza is not two 20 cm pizzas but four, and why large animals have disproportionate legs. Their mass grows as the cube while their bone cross-section grows only as the square. Another frequent trap: a triangle’s perimeter cannot be deduced from its base and height; infinitely many triangles share the same area with different perimeters. The tool flags this instead of displaying a number.