# Length of Arc of Astroid

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## Theorem

The total length of the arcs of an astroid constructed within a stator of radius $a$ is given by:

- $\mathcal L = 6 a$

## Proof

Let $H$ be embedded in a cartesian coordinate plane with its center at the origin and its cusps positioned on the axes.

We have that $\mathcal L$ is $4$ times the length of one arc of the astroid.

From Arc Length for Parametric Equations:

- $\displaystyle \mathcal L = 4 \int_{\theta \mathop = 0}^{\theta \mathop = \pi/2} \sqrt {\paren {\frac {\d x} {\d \theta} }^2 + \paren {\frac {\d y} {\d \theta} }^2} \rd \theta$

where, from Equation of Astroid:

- $\begin{cases} x & = a \cos^3 \theta \\ y & = a \sin^3 \theta \end{cases}$

We have:

\(\displaystyle \frac {\d x} {\d \theta}\) | \(=\) | \(\displaystyle -3 a \cos^2 \theta \sin \theta\) | |||||||||||

\(\displaystyle \frac {\d y} {\d \theta}\) | \(=\) | \(\displaystyle 3 a \sin^2 \theta \cos \theta\) |

Thus:

\(\displaystyle \sqrt {\paren {\frac {\d x} {\d \theta} }^2 + \paren {\frac {\d y} {\d \theta} }^2}\) | \(=\) | \(\displaystyle \sqrt {9 a^2 \paren {\sin^4 \theta \cos^2 \theta + \cos^4 \theta \sin^2 \theta} }\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 3 a \sqrt {\sin^2 \theta \cos^2 \theta \paren {\sin^2 \theta + \cos^2 \theta} }\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 3 a \sqrt {\sin^2 \theta \cos^2 \theta}\) | Sum of Squares of Sine and Cosine | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 3 a \sin \theta \cos \theta\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac {3 a \sin 2 \theta} 2\) | Double Angle Formula for Sine |

Thus:

\(\displaystyle \mathcal L\) | \(=\) | \(\displaystyle 4 \frac {3 a} 2 \int_0^{\pi / 2} \sin 2 \theta \rd \theta\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 6 a \intlimits {\frac {-\cos 2 \theta} 2} 0 {\pi / 2}\) | Primitive of $\sin a x$ | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 6 a \paren {-\frac {\cos \pi} 2 + \frac {\cos 0} 2}\) | evaluating limits of integration | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 6 a \frac {-\paren {-1} + 1} 2\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 6 a\) |

$\blacksquare$

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 11$: Special Plane Curves: Hypocycloid with Four Cusps: $11.11$ - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.21$: The Cycloid: Problem $7$