Gauss's magic shoelace area formula and its calculus companion

TL;DR

Learn about the shoelace formula, a simple method to calculate the area inside any convoluted curve made up of straight line segments.

Transcript

Welcome to another Mathologer video. The shoelace formula is a super simple way to calculate the exact area inside any convoluted curve made up of straight line segments, like my cat head curve over there. Even the great mathematician Carl Friedrich Gauss was impressed by this formula and mentioned it in his writings. The formula was certainly not ... Read More

Key Insights

• 🫥 The shoelace formula is a simple method to calculate the area inside convoluted curves made up of straight line segments.
• 😥 The formula involves filling in the coordinates of points on the curve, calculating the product of these coordinates, and summing them up.
• 😵 The formula can be explained by considering the triangles formed by the crosses in the formula, which cover the entire shape.
• 🔄 The sweeping direction of the radius affects the area calculated by the formula, with counterclockwise direction adding areas and clockwise direction subtracting areas.
• 😵 The formula does not work for self-intersecting curves, as the areas of the triangles formed by the crosses cannot be calculated correctly in such cases.
• 🫥 The shoelace formula can be used to approximate the area of curves that are not made up of straight line segments by calculating the area of a straight line approximation.
• ❓ The shoelace formula can be extended to calculate the exact area enclosed by complicated curves using calculus techniques and integral formulas.

Questions & Answers

Q: How does the shoelace formula work?

The shoelace formula involves filling in the coordinates of points on a curve and calculating the area using the product of these coordinates. The formula is based on the idea that the individual crosses in the formula correspond to triangles that cover the entire shape.

Q: Can the shoelace formula be used for curves that are not made up of straight line segments?

The shoelace formula can be used to approximate the area of curves that are not made up of straight line segments by calculating the area of a straight line approximation with points on the curve. By increasing the number of points, a closer approximation of the true area can be obtained.

Q: Why does the sweeping direction of the radius affect the area calculated by the shoelace formula?

When the sweeping direction of the radius changes from counterclockwise to clockwise, the shoelace formula subtracts the areas of certain triangles, resulting in a canceling effect. This ensures that the formula gives the correct area for convoluted curves.

Q: How can the shoelace formula be used for self-intersecting curves?

The shoelace formula does not work for self-intersecting curves because the areas of the triangles formed by the crosses cannot be calculated correctly. In such cases, the result of the formula may not represent the actual area enclosed by the curve.

Summary & Key Takeaways

• The shoelace formula is a method to calculate the exact area inside any convoluted curve made up of straight line segments.

• The formula involves filling in the coordinates of points on the curve and then using the product of these coordinates to calculate the area.

• The formula works for any closed curve in the xy-plane, as long as the curve does not intersect itself.