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The area enclosed by a circle as discussed in Euclidean geometry.
Strictly speaking, we are talking about the area of the disk whose boundary is the circle; however, the average person usually identifies the interior of a geometric shape with its boundary.
Depending on which circle constant you use, given a radius $r$ of a circle $\mathcal{C}$ in the Euclidean plane $\mathbb{R}^2$, the area of a circle is expressed either as $A(r) = \frac{1}{2} \tau r^2$ or as $A(r) = \pi r^2$.
In this proof, we are using the circle constant $\tau = 2 \pi$.
Given any Euclidean plane $\mathbb{R}^2$, one could select an orthonormal basis on $\mathbb{R}^2$ by postulating an origin $0$ at the center of the circle $\mathcal{C}$ and two orthonormal vectors $\hat{i}$ and $\hat{j}$. The circle $\mathcal{C}$ could be parameterized by a function $\overrightarrow{r}:[0, \tau] \times [0, r] \to \mathbb{R}^2$ defined as
Then the area of $\mathcal{C}$ is given by the following double integral?:
which evaluates to
In this proof, we are using the circle constant $\tau = 2 \pi$.
Given any Euclidean plane $\mathbb{R}^2$, one could select an orthonormal basis on $\mathbb{R}^2$ by postulating an origin $0$ at the center of the circle $\mathcal{C}$ and two orthonormal vectors $\hat{i}$ and $\hat{j}$. There is an geometric algebra $\mathbb{G}^2$ on the vector space defined by the equations $\hat{i}^2 = 1$, $\hat{j}^2 = 1$, and $\hat{i} \hat{j} = -\hat{j} \hat{i}$.
The circle $\mathcal{C}$ could be parameterized by a function $\overrightarrow{r}:[0, \tau] \to \mathbb{R}^2$ defined as
Then the area of $\mathcal{C}$ is given by integrating the magnitude of the areal velocity:
where $a \wedge b$ is the wedge product of two multivectors $a$ and $b$ and $\overrightarrow{v}$ is the velocity of a point in $\mathcal{C}$. This expression evaluates to
In this proof, we are using the circle constant $\tau = 2 \pi$.
Given any Euclidean plane $\mathbb{R}^2$, one could select an orthonormal basis on $\mathbb{R}^2$ by postulating an origin $0$ at the center of the circle $\mathcal{C}$ and two orthonormal vectors $\hat{i}$ and $\hat{j}$. The circle $\mathcal{C}$ could be parameterized by a function $\overrightarrow{r}:[0, \tau] \to \mathbb{R}^2$ defined as
Then the area of $\mathcal{C}$ is given by the action functional of the parameterized curve:
which evaluates to
In this proof, we are using the circle constant $\tau = 2 \pi$.
The area of a regular polygon $\mathcal{P}_n$ with $n$ sides and circumradius $r$ is given by the sequence of functions $P:\mathbb{N} \to (\mathbb{R} \to \mathbb{R})$
which embeds in the $\mathbb{R}_+$-action $A_\mathcal{P}^\prime:\mathbb{R}_+ \to (\mathbb{R} \to \mathbb{R})$, defined as
The limit of $A_\mathcal{P}^\prime$ as $n$ goes to infinity is the area of a circle with radius $r$:
Last revised on May 17, 2022 at 15:40:31. See the history of this page for a list of all contributions to it.