Geometric Area & Perimeter Calculator (All Shapes)

Basic Shapes

Rectangle: A = l × w. Perimeter: P = 2(l + w). Square: A = s². P = 4s. Diagonal: d = s√2. Triangle: A = ½ × base × perpendicular height. Heron's formula (3 sides): s = (a+b+c)/2, A = √(s(s−a)(s−b)(s−c)). Useful when height is unknown. Right triangle: P = a + b + √(a² + b²). Circle: A = πr². Circumference: C = 2πr = πd. Always use exact π for examination work — round at the end. These formulas form the basis of geometry — every more complex area can be decomposed into combinations of these.

Quadrilaterals

Parallelogram: A = base × perpendicular height. NOT base × slanted side. P = 2(a + b). Trapezium: A = ½ × (a + b) × h. Where a and b are the parallel sides, h is the perpendicular distance between them. P = sum of all 4 sides. Rhombus: special parallelogram with all sides equal. A = ½ × d₁ × d₂ (half product of diagonals). Kite: A = ½ × d₁ × d₂ (similar to rhombus). Cyclic quadrilateral (vertices on a circle): Brahmagupta's formula: A = √((s−a)(s−b)(s−c)(s−d)). All quadrilateral diagonals divide

Circles and Curved Shapes

Circle: A = πr². C = 2πr. Ellipse: A = π × a × b (where a, b are semi-major and semi-minor axes). Perimeter of ellipse has no closed-form solution. Approximation: P ≈ π[3(a+b) − √((3a+b)(a+3b))] (Ramanujan). Sector (pie slice): A = ½ r² × θ (θ in radians) or A = (θ/360°) × πr². Arc length: L = rθ (radians). Segment (area between chord and arc): A = ½ r²(θ − sin θ) (radians). Annulus (ring): A = π(R² − r²). Where R is outer radius, r is inner radius. Common in mechanical engineering for shafts an

Regular Polygons

Regular polygon (n equal sides of length s): A = ¼ × n × s² × cot(π/n). Apothem (distance from centre to side midpoint): a = s/(2 tan(π/n)). Circumradius (distance to vertex): R = s/(2 sin(π/n)). Examples: equilateral triangle (n=3): A = (√3/4) × s². Square (n=4): A = s². Pentagon (n=5): A ≈ 1.72 × s². Hexagon (n=6): A = (3√3/2) × s² ≈ 2.60s². As n increases, regular polygon approaches a circle. Circumscribed (around) and inscribed (inside) circles converge as n→∞. Used in architecture (regular