Welcome to our exploration of the area of a circle. A circle is a perfectly round shape where every point on the edge is exactly the same distance from the center. This distance is called the radius, represented by r. The area formula A equals pi r squared is one of the most important formulas in mathematics.
To understand how we derive the area formula, let's start by approximating the circle's area using rectangles. We can divide the circle into sectors and approximate each sector with a rectangle. The more rectangles we use, the closer our approximation becomes to the actual area of the circle.
Now let's see the key insight. If we cut the circle into many thin sectors and rearrange them alternately, we form a shape that looks like a rectangle. The width of this rectangle is half the circumference, which is pi r, and the height is the radius r. Therefore, the area equals pi r times r, giving us pi r squared.
Now let's see how the area formula works in practice. Starting with a circle of radius 1, the area is pi times 1 squared, which equals pi. When we double the radius to 2, the area becomes pi times 4, which is 4 times larger. Notice how the area increases with the square of the radius, not just proportionally.
The circle area formula appears everywhere in daily life. When ordering pizza, a 12-inch diameter pizza has much more area than an 8-inch one. In gardening, knowing the area helps calculate how much seed or fertilizer you need. The key insight is that area grows with the square of the radius, so doubling the radius gives you four times the area.