Shape velocity

Newton's Shapes Calculator

Compute geometry and physics quantities for shapes Newton often worked with (circle, ellipse, parabola, cycloid, hyperbola). Includes area, perimeter/arc-length and brachistochrone descent time for cycloid.

Choose shape Circle Ellipse Parabola (y = k x^2) Cycloid (brachistochrone) Hyperbola (right branch, x^2/a^2 - y^2/b^2 = 1)

Results

Enter parameters and click Compute.

Preview

Notes & formulas

Circle: area = π r^2, perimeter = 2 π r Ellipse: area = π a b, perimeter ≈ π[3(a+b) - sqrt((3a+b)(a+3b))] (Ramanujan) Parabola (y = k x^2): area between x0,x1 = ∫ y dx, arc length = ∫ sqrt(1+(2k x)^2) dx (numeric) Cycloid (parametric r, θ): x = r(θ - sinθ), y = r(1 - cosθ) Arc length from 0 to θ: S = 4 r [1 - cos(θ/2)] Brachistochrone descent time under gravity g from 0 to θ: T = sqrt(r/g) * θ Hyperbola (right branch): area not closed; arc length numeric between x0,x1