Euclidean Geometry of the Circle: Metric Relations, Calculus & Sector Topology
In Euclidean planar geometry, a circle is defined as the locus of all coplanar points equidistant from a fixed central point $O$. The constant distance between any point on the perimeter and the center is the radius ($r$). A circle represents the geometric plane figure possessing maximal area for a given perimeter (isoperimetric inequality) and complete continuous rotational symmetry of group $\text{SO}(2)$.
1. Fundamental Metric Relations & The Transcendental Constant $\pi$
The mathematical constant $\pi$ (pi) is defined as the invariant ratio of a circle's perimeter (circumference $C$) to its diameter ($d = 2r$):
$$\pi = \frac{C}{d} = \frac{C}{2r} \approx 3.141592653589793\dots$$
From this definition, the fundamental metric equations immediately follow:
- Circumference ($C$): $$C = 2\pi r = \pi d$$
- Radius from Circumference: $$r = \frac{C}{2\pi}$$
- Diameter from Circumference: $$d = \frac{C}{\pi}$$
2. Rigorous Derivation of Circular Surface Area ($A$)
The classic formula $A = \pi r^2$ can be rigorously derived through Archimedean polygon exhaustion or polar coordinate integral calculus. Dividing the circular disk into concentric differential rings of infinitesimal thickness $dr$ and perimeter $2\pi r$:
$$A = \int_0^r 2\pi \rho \, d\rho = 2\pi \left[ \frac{\rho^2}{2} \right]_0^r = \pi r^2$$
Expressed in terms of diameter $d$ and circumference $C$:
$$A = \frac{\pi d^2}{4} = \frac{C^2}{4\pi} = \frac{1}{2} C r$$
3. Circular Sector, Arc Length & Chord Trigonometry
A circular sector is a region bounded by two radii and an intercepted arc of central angle $\theta$:
A. Arc Length ($s$)
When angle $\theta$ is measured in radians ($\text{rad}$): $$s = r \cdot \theta$$ When angle $\alpha$ is measured in degrees ($^\circ$): $$s = 2\pi r \left( \frac{\alpha}{360^\circ} \right) = \frac{\pi r \alpha}{180^\circ}$$
B. Sector Area ($A_{\text{sector}}$)
In radians: $$A_{\text{sector}} = \frac{1}{2} r^2 \theta = \frac{1}{2} r s$$ In degrees: $$A_{\text{sector}} = \pi r^2 \left( \frac{\alpha}{360^\circ} \right)$$
C. Geometric Chord Length ($L_c$)
The straight-line segment connecting the two endpoints of an arc subtended by central angle $\theta$:
$$L_c = 2r \sin\left(\frac{\theta}{2}\right)$$
D. Circular Segment Area ($A_{\text{segment}}$)
The area between a chord and its arc equals the sector area minus the area of the isosceles triangle formed by the radii and chord:
$$A_{\text{segment}} = A_{\text{sector}} - A_{\text{triangle}} = \frac{1}{2} r^2 (\theta - \sin\theta) \quad (\theta \text{ in radians})$$
4. Concentric Annulus (Circular Ring) Geometry
An annulus is the planar region enclosed between two concentric circles of outer radius $R$ and inner radius $r$ ($R > r$):
$$A_{\text{annulus}} = \pi R^2 - \pi r^2 = \pi (R^2 - r^2) = \pi (R - r)(R + r)$$
By the Pythagorean theorem, if a chord of the outer circle is tangent to the inner circle and has length $2c$, the annular area simplifies to:
$$A_{\text{annulus}} = \pi c^2$$