Geometry formulas you actually use
Eleven shapes that cover most of school geometry, with the formulas that turn dimensions into answers.
Why these eleven shapes?
A glance at any school exam paper or any engineering reference reveals the same small cast of geometric shapes appearing again and again. Rectangles and triangles dominate 2D problems; cuboids, cylinders, and spheres dominate 3D. Once you can compute area, perimeter, surface area, and volume for these eleven, you can handle the great majority of practical problems.
2D shapes
Rectangle. Area = length × width. Perimeter = 2(length + width). The diagonal is √(l² + w²) by Pythagoras.
Square. A rectangle with all sides equal. Area = side². Perimeter = 4 × side. Diagonal = side × √2.
Triangle. The basic formula is Area = ½ × base × height. For any triangle whose three sides you know, Heron’s formula computes the area without needing the height: A = √(s(s−a)(s−b)(s−c)) where s = (a + b + c)/2.
Circle. Area = πr². Circumference = 2πr. Diameter = 2r. The most efficient shape: it has the smallest perimeter for a given area.
Trapezoid. Area = ½ × (a + b) × h, where a and b are the two parallel sides and h is the perpendicular distance between them.
3D shapes
Cube. Volume = side³. Surface area = 6 × side². Space diagonal = side × √3.
Cuboid (rectangular box). Volume = length × width × height. Surface area = 2(lw + lh + wh). Space diagonal = √(l² + w² + h²).
Sphere. Volume = (4/3)πr³. Surface area = 4πr². The most efficient 3D shape: it minimises surface area for a given volume, which is why bubbles, planets, and water droplets gravitate toward it.
Cylinder. Volume = πr² × height. Surface area = 2πr(r + h), of which 2πrh is the side and 2πr² is the two end caps.
Cone. Volume = (1/3)πr² × height. The slant height is √(r² + h²). Surface area = πr(r + slant), one circular base plus the curved lateral surface.
Square pyramid. Volume = (1/3) × base² × height. The slant height is √((base/2)² + h²), and the surface area is base² + 2 × base × slant.
Why volume scales as length cubed
If you double every linear dimension of any shape, the perimeter doubles, the area quadruples, and the volume goes up by a factor of eight (2³). This sounds obvious but has dramatic real-world consequences. Bigger animals have proportionally more volume than surface area, so they retain heat better; this is why mice freeze quickly and elephants overheat in heatwaves. The same scaling explains why scaled-up insects from B-movies could not actually walk: their leg cross-section grows as length², but their weight grows as length³, so beyond a certain size their legs would snap.
Quick problem-solving sequence
- Draw the shape and label every length you know.
- Identify the formula you need (area, perimeter, surface area, or volume).
- Plug numbers in carefully, paying attention to units.
- Sanity-check the answer's units and order of magnitude.
If you would rather just type in dimensions, the geometry calculator handles all eleven shapes.