Welche Kurve ist länger: BLAU oder ROT? 🤔

Summary

The video compares the lengths of a blue curve and a red curve, each made of semicircular arcs. Both curves have the same total length.

Method

Red curve

The red curve consists of two semicircles with radii (x) and (y). A semicircle’s curved length is half the circumference of a full circle:

[ \frac{1}{2}(2\pi r)=\pi r ]

Therefore, the red curve’s length is:

[ L_{\text{red}}=\pi x+\pi y ]

Blue curve

The blue curve is one semicircle whose diameter spans the two smaller diameters, (2x) and (2y). Its diameter is:

[ 2x+2y ]

So its radius is (x+y), and its curved length is:

[ L_{\text{blue}}=\pi(x+y) ]

Comparison

Expanding the blue curve’s length gives:

[ L_{\text{blue}}=\pi x+\pi y ]

This matches the red curve’s length, so neither curve is longer; they are equal in length.

Speakers and sources

• Presenter: Susanne, identified in the video description as a presenter from the MathemaTrick channel.
• Other speakers or sources: None featured.