📐 Geometry Calculators

Area Formulas for Common Shapes

Area measures the amount of two-dimensional space enclosed by a shape, expressed in square units. The most important area formulas: rectangle = length × width; triangle = ½ × base × height (where height is perpendicular to the base); circle = π × r²; trapezoid = ½ × (a + b) × h where a and b are the parallel sides; parallelogram = base × height. The area of a regular polygon with n sides of length s is A = (n × s²) ÷ (4 × tan(π/n)). For irregular shapes, the shoelace formula computes area from vertex coordinates: A = ½|Σ(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)|. Area scales with the square of linear dimensions — doubling all lengths quadruples the area.

The Pythagorean Theorem

In a right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of squares of the other two sides: a² + b² = c². Pythagorean triples are integer sets satisfying this equation — the smallest is (3, 4, 5): 9 + 16 = 25. Multiples of triples also work: (6, 8, 10), (5, 12, 13), (8, 15, 17). The theorem has hundreds of proofs and extends to 3D: the space diagonal of a rectangular box with dimensions a, b, c is √(a² + b² + c²). The converse is equally useful — if a² + b² = c² holds for the three sides of a triangle, the triangle contains a right angle at C. This is used in construction to verify that corners are square.

Circle Properties

A circle is defined as all points equidistant from a centre. Its properties: diameter = 2r, circumference = 2πr = πd, area = πr². The ratio of circumference to diameter is π ≈ 3.14159, an irrational and transcendental number. A sector (pie slice) with central angle θ (in degrees) has arc length = (θ/360) × 2πr and sector area = (θ/360) × πr². A chord divides the circle into two segments; the perpendicular from the centre to any chord bisects that chord. The inscribed angle theorem states that an inscribed angle is always half the central angle subtending the same arc — a theorem with direct applications in optics and mechanical design.

Volume of 3D Shapes

Volume measures three-dimensional space, expressed in cubic units. Key formulas: cube = s³; rectangular prism = l × w × h; cylinder = πr²h; cone = ⅓πr²h; sphere = (4/3)πr³; pyramid = ⅓ × base area × height. Notice that a cone has exactly one-third the volume of a cylinder with the same base and height, and a pyramid has one-third the volume of a prism — a surprising and elegant relationship. Volume scales with the cube of linear dimensions: doubling all lengths multiplies volume by 8. The surface area to volume ratio decreases as objects get larger, which has profound biological implications — it explains why small animals lose body heat rapidly and why large animals need more efficient cooling systems.