Welcome! Today we'll discover how to find the area of a circle using a clever triangle method. Imagine a circle is made up of many thin rings, like layers of an onion. When we unroll and stack these rings, something amazing happens - they form a triangle!
Now let's see what happens when we unroll each ring. Each ring becomes a straight line segment. The innermost ring becomes the shortest line, while the outermost ring becomes the longest line, with length equal to the circumference two pi r.
Now comes the magic! When we stack these unrolled lines from shortest to longest, they form a triangle. The height of this triangle equals the radius r, and the base equals the circumference two pi r. This triangle approximates the area of our original circle.
Now we can calculate the area using the triangle formula. The area of a triangle is one half times base times height. Substituting our values: one half times two pi r times r. This simplifies to pi r squared - the famous formula for the area of a circle!
To summarize what we've learned: A circle can be thought of as many thin rings. When we unroll these rings and stack them, they form a triangle. Using the triangle area formula with height r and base two pi r, we derive the circle area formula pi r squared. This beautiful method shows the geometric connection between circles and triangles.