Area of a Circle

Using the definition of area, we can calculate the sum of unit squares for any object with a closed space. In the case of a circle, a circle has a defined radius and an angle radians of 2π.

In the simulation below, you can input an arbitrary radius of your choice. Note that there are small unit squares on the graph, which can be counted manually to get the area of the circle. Although we would get the correct answer, this is not practical as a circle with a large enough radius would take a long time to count up all the individual unit squares each of length and width 1m by 1m.

Radius:

Since a circle is a two dimensional shape and to plot the radius and angle require two dimensions, we can assume that a circle will be in two dimensions. Since every circle has an angle radians of 2π and a radius r, we can take the integral of the product of these values with respect to the radius.

\[ A = \int_{0}^{r} 2 \pi r dr \]

We integrate from 0 to r, that is from a circle that has a radius of 0 to an arbitrary radius r.

\[ A = \int_{0}^{r} 2 \pi \frac{r^2}{2} \]

\[ A = 2 \pi \frac{r^2}{2} - 2 \pi \frac{0^2}{2} \]

\[ A = 2 \pi \frac{r^2}{2} - 0 \]

\[ A = 2 \pi \frac{r^2}{2} \]

\[ A = \pi r^2 \]

Since a circle's angle is not factored into calculating the area of a circle, you would just input the radius.

Therefore the equation we get to calculate the area of a circle is π r².

\[ A = \pi \times r^2 \]

A circle has a radius that goes 360 degrees, and is an enclosed space, on the xy plane in Cartesian coordinates, or (r, theta) plane in polar coordinates, so that would mean that the units of a circle's area would be m².

\[ m^2 \]

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