3D Shapes — Volume & Surface Area Calculator

3D Shapes Formulas Guide

Cube and Cuboid

Cube (side length a): Volume = a³. Surface area = 6a². All faces identical squares. Cuboid (length l, width w, height h): Volume = l × w × h. Surface area = 2(lw + lh + wh) — 2 of each of the three face types. Diagonal of cuboid = √(l² + w² + h²). Applications: volume of water in a rectangular tank, material needed for a rectangular block, space in a room (cuboid air volume).

Sphere and Cylinder

Sphere (radius r): Volume = (4/3)πr³. Surface area = 4πr². Hemisphere: Volume = (2/3)πr³. Curved surface area = 2πr². Total area = 3πr². Cylinder (radius r, height h): Volume = πr²h. Curved surface area = 2πrh. Total surface area = 2πr(r + h). The 2πr(r+h) formula is elegant: unroll the curved surface into a rectangle (2πr × h) and add two circles (πr² each).

Cone and Pyramid

Cone (radius r, height h): Slant height l = √(r² + h²). Volume = (1/3)πr²h. Curved surface area = πrl. Total surface area = πr(r + l). Cone volume is exactly 1/3 of the cylinder with the same base and height — place a cone inside a cylinder and fill with water: 3 cone-fills = 1 cylinder-fill. Square pyramid (base side a, height h): Slant height l = √((a/2)² + h²). Volume = (1/3)a²h. Surface area = a² + 2al. The great pyramids of Giza: height 146.5m, base 230m. Volume ≈ (1/3) × 230² × 146.5 ≈ 2.5

Composite Shapes and Applications

Many real objects are combinations of basic 3D shapes. A silo = cylinder + hemisphere cap. A room with a bay window = cuboid + triangular prism. A tent = triangular prism (rectangular floor plan) or cone (circular base). Strategy for composite shapes: break into components, calculate volume/area of each, add volumes, and for surface area — add only the exposed surfaces (internal interfaces between components are not surface area of the composite). Applied: volume of concrete needed for a cylindr