# Area of Circle

## Contents

## Theorem

The area $A$ of a circle is given by:

- $A = \pi r^2$

where $r$ is the radius of the circle.

## Proof 1

From Equation of Circle:

- $x^2 + y^2 = r^2$

Thus $y = \pm \sqrt {r^2 - x^2}$.

It follows that from the geometric interpretation of the definite integral:

\(\displaystyle A\) | \(=\) | \(\displaystyle \int_{-r}^r \left({\sqrt {r^2 - x^2} - \left({-\sqrt {r^2 - x^2} }\right)}\right) \rd x\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \int_{-r}^r 2 \sqrt {r^2 - x^2} \rd x\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \int_{-r}^r 2 r \sqrt {1 - \frac {x^2} {r^2} } \rd x\) | $\quad$ | $\quad$ |

Let $x = r \sin \theta$ (note that we can do this because $-r \le x \le r$).

Thus $\theta = \arcsin \left({\dfrac x r}\right)$ and $\rd x = r \cos \theta \rd \theta$.

\(\displaystyle A\) | \(=\) | \(\displaystyle \int_{\arcsin \left({\frac {-r} r}\right)}^{\arcsin \left({\frac r r}\right)} 2 r^2 \sqrt {1 - \frac {\left({r \sin \theta}\right)^2} {r^2} } \cos \theta \rd \theta\) | $\quad$ Integration by Substitution | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \int_{-\frac \pi 2}^{\frac \pi 2} 2 r^2 \sqrt {1 - \sin^2 \theta} \cos \theta \rd \theta\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \int_{-\frac \pi 2}^{\frac \pi 2} 2 r^2 \sqrt {\cos^2 \theta} \cos \theta \rd \theta\) | $\quad$ Sum of Squares of Sine and Cosine | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle r^2 \int_{-\frac \pi 2}^{\frac \pi 2} 2 \cos^2 \theta \rd \theta\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle r^2 \int_{-\frac \pi 2}^{\frac \pi 2} \left({1 + \cos \left({2 \theta}\right)}\right) \rd \theta\) | $\quad$ Double Angle Formula for Cosine: Corollary 1 | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle r^2 \left[{\theta + \frac 1 2 \sin \left({2 \theta}\right)}\right]_{-\frac \pi 2}^{\frac \pi 2}\) | $\quad$ from Integration of Constant and Primitive of Cosine Function | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle r^2 \left[{\frac \pi 2 + \frac 1 2 \sin \left({2 \cdot \frac {-\pi} 2}\right) - \frac {-\pi} 2 - \frac 1 2 \sin \left({2 \cdot \frac \pi 2}\right)}\right]\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle r^2 \left[{2 \cdot \frac \pi 2 + 2 \cdot \frac 1 2 \cdot 0}\right]\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \pi r^2\) | $\quad$ | $\quad$ |

$\blacksquare$

## Proof 2

Proof by shell integration:

The circle can be divided into a set of infinitesimally thin rings, each of which has area $2 \pi t \ \mathrm dt$, since the ring has length $2 \pi t$ and thickness $\mathrm d t$.

\(\displaystyle A\) | \(=\) | \(\displaystyle \int_0^r 2\pi t \ \mathrm d t\) | $\quad$ Perimeter of Circle | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \left[{\pi t^2}\right]_0^r\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \pi r^2\) | $\quad$ | $\quad$ |

$\blacksquare$

## Proof 3

Refer to the figure.

Construct a circle with radius r and circumference $c$, where its area is denoted by $C$.

Construct a triangle with height r and base $c$, where its area is denoted by $T$.

### Lemma 1: $T \mathop = \pi r^2$

\(\displaystyle T\) | \(=\) | \(\displaystyle \frac{rc}2\) | $\quad$ Area of Triangle in Terms of Side and Altitude | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac{r 2 \pi r} 2\) | $\quad$ Perimeter of Circle | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \pi r^2\) | $\quad$ | $\quad$ |

$\Box$

### Lemma 2: $T \ge C$

This will be proven by contradiction.

Assume $T < C$.

It should be possible to construct a regular polygon with area $P$, where $T < P < C$.

For any given regular polygon:

- $P = \dfrac {hq} 2$

where:

- $q$ is the perimeter of the polygon
- $h$ is the height of any given triangular part of it
- $P$ is the area.

On one hand:

- $P > T \implies \dfrac {hq} 2 > \dfrac {rc} 2$

On the other hand:

- $0 < h < r \land 0 < q < c \implies \dfrac {hq} 2 < \dfrac {rc} 2$

Hence a contradiction is obtained.

Hence $\neg T < C$.

Hence $T \ge C$.

$\Box$

### Lemma 3: $T \le C$

This will be proven by contradiction.

Assume $T > C$.

It should be possible to construct a regular polygon with area $P$, where $C < P < T$.

From Area of Polygon by Inradius and Perimeter:

- $P = \dfrac {hq} 2$

where:

- $q$ is the perimeter of the regular polygon
- $h$ is the inradius of the regular polygon
- $P$ is the area.

as each triangle has the base $B = \dfrac {q} n$ and area $A = \dfrac {hq}{2n}$ and with $n$ triangles we get $P = \dfrac {hq} 2$

On one hand:

- $P < T \implies \dfrac {hq} 2 < \dfrac {rc} 2$

On the other hand:

- $0 < h = r \land 0 < c < q \implies \dfrac {hq} 2 > \dfrac {rc} 2$

Hence a contradiction is obtained.

Hence $\neg T > C$.

Hence $T \le C$.

$\Box$

### Final Proof

- $T \ge C$ (from Lemma 2)
- $T \le C$ (from Lemma 3)
- $\therefore T \mathop = C$
- $\therefore C \mathop = T \mathop = \pi r^2$ (from Lemma 1)

$\blacksquare$

## Proof 4

Expressing the area in polar coordinates:

\(\displaystyle \iint \, \mathrm d A\) | \(=\) | \(\displaystyle \int_0^r \int_0^{2 \pi} t \, \mathrm d t \, \mathrm d \theta\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \left.{\int_0^r t \theta }\right \vert_0^{2 \pi} \, \mathrm d t\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \int_0^r 2 \pi t \, \mathrm d t\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 2 \pi \left({\left.{\frac 1 2 t^2}\right\vert_0^r}\right)\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 2 \pi \left({\frac 1 2 r^2}\right)\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \pi r^2\) | $\quad$ | $\quad$ |

$\blacksquare$

## Proof 5

Let the circle of radius $r$ be divided into many sectors:

If they are made small enough, they can be approximated to triangles whose heights are all $r$.

Let the bases of these triangles be denoted:

- $b_1, b_2, b_3, \ldots$

From Area of Triangle in Terms of Side and Altitude, their areas are:

- $\dfrac {r b_1} 2, \dfrac {r b_2} 2, \dfrac {r b_3} 2, \ldots$

The area $\mathcal A$ of the circle is given by the sum of the areas of each of these triangles:

\(\displaystyle \mathcal A\) | \(=\) | \(\displaystyle \dfrac {r b_1} 2 + \dfrac {r b_2} 2 + \dfrac {r b_3} 2 + \cdots\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \dfrac r 2 \left({b_1 + b_2 + b_3 + \cdots}\right)\) | $\quad$ | $\quad$ |

But $b_1 + b_2 + b_3 + \cdots$ is the length of the circumference of the circle.

From Perimeter of Circle:

- $b_1 + b_2 + b_3 + \cdots = 2 \pi r$

Hence:

\(\displaystyle \mathcal A\) | \(=\) | \(\displaystyle \dfrac r 2 \left({b_1 + b_2 + b_3 + \cdots}\right)\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \dfrac r 2 \left({2 \pi r}\right)\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \pi r^2\) | $\quad$ | $\quad$ |

It needs to be noted that this proof is intuitive and non-rigorous.

$\blacksquare$

## Proof 6

From Equation of Circle:

- $x^2 + y^2 = r^2$

Let $A$ be the area of the circle whose equation is given by $x^2 + y^2 = r^2$.

We have that:

- $y = \pm \sqrt {r^2 - x^2}$

For the upper half of the circle:

- $y = +\sqrt {r^2 - x^2}$

Thus for the right hand half of the upper half of the circle:

\(\displaystyle \frac A 4\) | \(=\) | \(\displaystyle \int_0^r \sqrt {r^2 - x^2} \rd x\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac {\pi r^2} 4\) | $\quad$ Definite Integral from $0$ to $r$ of $\sqrt {r^2 - x^2}$ | $\quad$ |

Hence the result.

$\blacksquare$

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 4$: Geometric Formulas: $4.11$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41971 \ldots$