Engaging Geometry: Exploring the Half Fish Problem
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Chapter 1: Introduction to the Half Fish Puzzle
I want to extend my gratitude to John Clark Craig for sharing today’s mathematical challenge. This puzzle resembles a fish that has been sliced in half! The area, denoted as A, is the overlap of two quarter circles, each with a radius of 2, while the entire figure is contained within a square of side length 2.
Before continuing, I encourage you to pause and try solving it on your own with some pen and paper. Once you’re ready, proceed to find the solution!
Section 1.1: Forming the Inner Triangle
To tackle the problem, let’s first visualize an inner triangle within area A.
The base of this triangle matches the length of the square’s side, which is 2. The other two sides correspond to the radii of the quarter circles, also measuring 2. This configuration creates an equilateral triangle!
Subsection 1.1.1: Calculating the Triangle's Area
Now, we need to determine the area of this triangle. Given that each angle measures 60 degrees, we can apply the formula 1/2(a)(b)sin(c).
Let’s explore how to find the area of A. Visualize this: the area A can be calculated as the sum of two 60-degree sectors minus the area of the triangle.
One sector contributes to the area, along with a small wedge. Therefore, two sectors yield twice that contribution plus two small wedges. If we subtract one sector, we are left with two small wedges and a larger area, which is what A represents.
Mathematically, we can express it as follows:
The fraction 1/6 indicates that the sector measures 60 degrees, as 60/360 simplifies to 1/6. And that leads us to our conclusion!
Bonus Question: Area of the Fin?
Can you also determine the area of the fish's fin?
John’s solution reveals a fascinating number: 2.4567, which is intriguingly close to 2.34567. How interesting!
What was your thought process while solving this? I’d love to hear your insights in the comments below!
Section 1.2: Additional Resources
Here are some engaging math puzzles you might enjoy:
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