What is the element equation for a chord approximating an arc in the finite element method?

Conditions

• The element is modeled as a straight chord of a circle.
• $r$ is the circle radius.
• $\theta$ is the central angle subtended by the element.

Reasoning, step by step

1. Identify the radius $r$ and the half central angle $\theta/2$ from the diagram.
2. Use the right-triangle relation to find the length of one half of the chord, which is $r\sin(\theta/2)$.
3. Combine the two equal halves of the chord to get the full element length.
4. The resulting element equation is $2r\sin\left(\frac{\theta}{2}\right)$.

Example

The video shows a diagram with a red triangle where the radius is labeled $r$, the half-angle is $\theta/2$, and the chord segments are labeled $r\sin(\theta/2)$. The speaker states the height can be computed as “two times radius times sine of theta by two.”

Common misconceptions

• Confusing the central angle $\theta$ with the half-angle $\theta/2$ in the sine term.
• Assuming the element equation represents the arc length rather than the straight chord length.

Watch the explanation

Explore next

• How does the sum of element lengths relate to the approximated circle?
• What is discretization error in this context?
• Why are element shape functions needed?