Geometry Calculator
Calculate area and perimeter for 2D shapes (circles, rectangles, triangles, trapezoids, sectors) and volume and surface area for 3D solids (cubes, prisms, cylinders, spheres, cones).
Frequently Asked Questions
What are the most important area formulas to know?
Key 2D area formulas: Rectangle = length × width. Square = side². Triangle = ½ × base × height. Circle = π × r² (π ≈ 3.14159). Parallelogram = base × height. Trapezoid = ½ × (base₁ + base₂) × height. Regular hexagon = (3√3/2) × side². Sector (pie slice) = (angle/360) × π × r². For any irregular polygon, split into triangles. Heron's formula gives triangle area from three sides: Area = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2.
What is the difference between area and perimeter?
Perimeter is the total distance around the outside edge of a 2D shape (measured in linear units: cm, m, ft). Area is the total surface enclosed by that boundary (measured in square units: cm², m², ft²). For a circle, perimeter is called circumference (C = 2πr or πd) and area = πr². A practical distinction: perimeter determines fencing needed around a garden; area determines turf needed to fill it. Increasing dimensions proportionally: doubling all sides quadruples area but only doubles perimeter.
What are the volume and surface area formulas for common 3D shapes?
Key 3D formulas: Cube — Volume = s³; Surface area = 6s². Rectangular prism (box) — V = lwh; SA = 2(lw + lh + wh). Cylinder — V = πr²h; SA = 2πr² + 2πrh. Sphere — V = (4/3)πr³; SA = 4πr². Cone — V = (1/3)πr²h; SA = πr² + πrl (where l = slant height = √(r²+h²)). Triangular prism — V = triangle area × length; SA = sum of all 5 faces. Volume is in cubic units; surface area in square units.
How do geometry formulas apply to real-world problems?
Geometry calculations are fundamental to many everyday and professional tasks: Flooring/tiling: area of room ÷ area per tile = number of tiles needed (add 10% for waste). Painting: wall area ÷ coverage rate (m²/litre) = litres of paint needed. Fencing: perimeter of yard = metres of fencing. Plumbing: pipe cross-section area = π × r² determines flow capacity. Construction: concrete volume for a cylinder footing = πr²h. Land surveying: irregular plots are divided into triangles and summed. Always use consistent units — mixing metres and centimetres is the most common calculation error.
What is π (pi) and why does it appear in circle and sphere calculations?
Pi (π ≈ 3.14159265…) is the ratio of any circle's circumference to its diameter — a constant that is the same for every circle in the universe. It is irrational (never terminates or repeats as a decimal) and transcendental (not a root of any polynomial with rational coefficients). Pi appears whenever a calculation involves circular or spherical symmetry: wheel rotations, wave frequencies, pendulum motion, normal distributions in statistics (Bell curve formula contains e^(−x²) and divides by √(2π)), and Einstein's field equations in general relativity.