Calculating Circle Area

The formula to calculate a circle's area (A) is $$ A = \pi r^2 $$ where \(r\) stands for the circle's radius, and $ \pi $ is the constant pi, a fundamental mathematical constant.

The symbol \(\pi\) (pi) represents pi, approximately 3.14159. For simplicity in calculations, it's often rounded to 3.14.

An Illustrative Example

Let's look at a circle with a radius (r) of 2.2361.

To find the area of this circle, we apply the formula:

$$ A = \pi r^2  = 15.708 $$

Thus, the circle's area is approximately 15.708 square units.

Verification: For confirmation, the area was also calculated using GeoGebra, yielding the same result.

Demonstration

Consider a circle with a radius r and a circumference c.

It has been established that a circle's area is equivalent to the area of a triangle with the circumference as its base and the radius as its height.

With this equivalence in mind, the area of the circle equals the triangle's area, calculated as half the product of its base and height:

$$ A_c = \frac{1}{2} \cdot \text{base} \cdot \text{height} $$

Here, the base is the circumference (c), and the height is the radius (r).

$$ A_c = \frac{1}{2} \cdot c \cdot r $$

Knowing the circumference is $ c = 2 \pi r $,

$$ A_c = \frac{1}{2} \cdot (2 \pi r ) \cdot r $$

$$ \require{cancel} A_c = \pi r^2 $$

This derivation confirms the formula for calculating a circle's area.

And so forth.