Eclipsed Geometry: Solving a Semicircle Puzzle
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Chapter 1: Introduction to the Puzzle
This intriguing figure resembles a red moon overshadowed by smaller celestial bodies in an alternate universe. What are your thoughts on this visual?
The challenge features three semicircles, each with a radius of 1, situated on the diameter AB of a larger semicircle with a radius of 2. The centers of the smaller semicircles equally segment AB into four parts.
Before you continue, take a moment to grab some paper and a pencil. Give the problem a shot before uncovering the solution!
Solution Overview
Geometry often demands creative thinking. In this solution, we will draw six lines to segment the smaller semicircles further.
Section 1.1: Analyzing the Structure
The smaller semicircles consist of five congruent sectors and two equilateral triangles. The light blue shapes represent the triangles.
Now, our objective is to calculate the total area of the figure along with the individual areas of the sectors and triangles.
Subsection 1.1.1: Total Area Calculation
The overall area corresponds to that of a semicircle with a radius of 2.
Section 1.2: Areas of Individual Components
For each sector, the angle measures 60 degrees due to their congruence. Thus, we deduce that the area of each sector is one-sixth of the total semicircle area.
Next, let’s examine the triangles. Each triangle is equilateral, with angles of 60 degrees. We employ the area formula ( A = frac{1}{2}(a)(b)sin(c) ).
Where ( a = b = 1 ) (the sides of the triangle) and ( c = sin(60) = frac{sqrt{3}}{2} ).
To find the area of the red region, we subtract the areas of the two triangles and the five sectors from the total figure.
Mathematically, this can be expressed as:
And that's our final answer!
Chapter 2: Video Resources
For further insights into this geometric problem, check out these informative videos!
The first video, Finding Area of a Shaded Region in a Semicircle, provides a visual demonstration to aid understanding of the area calculations.
In the second video, Can You Find the Red Area of This Semicircle? A Nice Geometry Problem Test Your Math Skills Part 33, viewers are challenged to solve similar problems and enhance their math skills.
Conclusion
How fascinating is this geometry puzzle? What thoughts crossed your mind while solving it? Feel free to share your insights in the comments below!
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