EQUATION

Large-Angle Pendulum Period (Exact)

Gives the exact period of a simple pendulum at any amplitude via the complete elliptic integral of the first kind: T = 4·√(L/g)·K(sin(θ₀/2))

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The equation

EQ.LARGE-ANGLE-PENDULUM-PERIOD

T = 4\sqrt{\frac{L}{g}}\,K\!\left(\sin\frac{\theta_0}{2}\right)

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What it solves

Gives the exact period of a simple pendulum at any amplitude via the complete elliptic integral of the first kind: T = 4·√(L/g)·K(sin(θ₀/2)). This reduces to the small-angle formula when θ₀ → 0.

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When to use it

When amplitude is large (typically > 30°–40°) and precision matters. The elliptic integral is easily evaluated numerically using the AGM algorithm or its series expansion.

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When NOT to use it

Overkill for small angles (θ₀ < 15°) where T = 2π√(L/g) is already accurate to 0.5%. Also assumes a simple (point-mass) pendulum; physical pendulums need the moment-of-inertia form.

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Common mistakes

Confusing the modulus k = sin(θ₀/2) with the amplitude θ₀ itself. Using K(θ₀/2) instead of K(sin(θ₀/2)). Forgetting the prefactor 4√(L/g) and writing 2π√(L/g)·K(k) instead.

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Topics that use this equation

beyond small angles