Calculate exact volumes, surface areas, and capacities for cylinders, cones, cubes, prisms, pyramids, ellipsoids, tori, and Platonic polyhedra.
Calculate the volume of a regular hexahedron (cube) from edge length with step-by-step arithmetic working and units conversion.
Calculate total exterior surface area of a cube with formula A = 6s² across metric and imperial measurement units.
Calculate rectangular prism volume from length, width, and height with step-by-step arithmetic and units conversion.
Calculate total outer surface area of a rectangular prism from length, width, and height with algebraic steps.
Calculate cylinder volume from circular base radius and perpendicular height with step-by-step calculations and fluid conversion.
Calculate lateral and total surface area of a cylinder with formula A = 2πr(h + r). Includes worked steps and base area breakdowns.
Calculate the volume of a right circular cone with formula V = 1/3πr²h. Step-by-step arithmetic working and cylinder ratio comparison.
Calculate slant height, lateral area (πrs), and total closed surface area of a cone with step-by-step Pythagorean working.
Calculate the volume of square, rectangular, and regular pyramids from base dimensions and vertical apex height.
Calculate the volume of any uniform cross-section prism from cross-sectional base area and length/height.
Simultaneously calculate volume, surface area, and 3D space diagonal for any rectangular prism with full working.
Calculate triangular prism volume from triangular cross-section base, height, and extruded length with step solutions.
Universal hemisphere solver: compute volume, curved surface area, and total surface area simultaneously from radius.
Calculate the volume of a triaxial ellipsoid or spheroid from semi-principal axes a, b, and c with step proofs.
Calculate volume and surface area of a doughnut-shaped torus using Pappus's Centroid Theorem with major and minor radii.
Compare cylinder and sphere volumes. Archimedes 2:3 ratio theorem, packing efficiency, and manufacturing trade-offs.
Understand the geometric relationship between cone and sphere volumes. Proportions, formulas, and sandpile geometry.
Calculate cylinder volume with formula V = pi r^2 h. Step-by-step arithmetic working, circular base area, and liquid tank capacity.
Step-by-step cone volume guide. Formula V = 1/3 pi r^2 h, finding height from slant height using Pythagorean theorem, and worked examples.
Calculate cube volume from side length. Formula V = a^3, surface area A = 6a^2, space diagonal length, and step-by-step arithmetic examples.
Calculate rectangular box volume. Length times width times height formula, surface area calculations, and shipping box capacity.
Learn how to find triangular prism volume. Triangle base area times length formula V = 1/2 b h L with clear arithmetic steps.
Calculate pyramid volume from base length and vertical height. Formula V = 1/3 a^2 h, Great Pyramid of Giza dimensions, and solved steps.
Learn the 3D stretched sphere volume formula V = 4/3 pi a b c. Semi-axes explanation, Earth oblate spheroid volume, and calculus derivation.
Calculate torus (donut) volume and surface area using Pappus Centroid Theorem. Formulas V = 2 pi^2 R r^2 and A = 4 pi^2 R r.
Calculate capsule volume for pharmaceuticals and pressure vessels. Formula V = pi r^2 (4/3 r + h) combining a cylinder with two hemispheres.
Calculate truncated cone (frustum) volume. Formula V = 1/3 pi h (R^2 + r^2 + R r), bucket capacity, and lampshade geometry.
Calculate regular octahedron volume from edge length a. Formula V = (sqrt(2)/3) a^3, Platonic solids symmetry, and diamond crystallography.
Calculate regular tetrahedron volume. Formula V = a^3 / (6 sqrt(2)), height formula h = a sqrt(2/3), and methane molecular geometry.
Learn how to measure volume for irregular shapes. Water displacement, mass-density conversion, 3D laser photogrammetry, and Simpson numerical integration.
Step-by-step guide to measuring volume using graduated cylinders. Meniscus reading, buoyant objects handling, and density calculation.
Master the density triangle: Density = Mass / Volume. Rearranging for volume V = m / rho, unit conversions (g/cm3 to kg/m3), and material tables.
Learn how buoyant force depends on displaced sphere volume. Formula F_b = rho V g, neutral buoyancy submarines, and weather balloons.
Calculate storage tank capacity. Liquid dipstick readings for horizontal cylinders, spherical Horton spheres, and vertical fuel tanks.
Calculate Earth volume. Mean volumetric radius 6,371 km vs equatorial bulging ellipsoid WGS84 measurements.
