Geometric Area & Perimeter Calculator (All Shapes)

What's the key thing to understand about 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.

What's the key thing to understand about 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 the shape into triangles, which is actually the basis for deriving most of these area formulas in the first place.

What's the key thing to understand about 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 and pipes, where the annulus formula calculates the cross-sectional area of material in a hollow cylindrical component.

What do I need to know about 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, where regular polygons appear in floor plans, window designs, and structural elements chosen partly for their geometric elegance and partly for their load-bearing efficiency.