Circle Calculator

Circle Geometry Formulas & Derivations

A circle is defined as the set of all points in a two-dimensional plane that are at an equal distance (the radius $r$) from a central point. The fundamental relationships governing circle geometry include:

• Diameter: $d = 2r$ (The longest chord passing through the center).
• Circumference: $C = 2\pi r = \pi d$ (The perimeter or boundary distance around the circle).
• Area: $A = \pi r^2 = \frac{\pi d^2}{4} = \frac{C^2}{4\pi}$.
• Arc Length ($s$): $s = r \times \theta_{\text{rad}} = 2\pi r \times \left(\frac{\theta^\circ}{360^\circ}\right)$.
• Sector Area ($A_{\text{sector}}$): $A_{\text{sector}} = \frac{1}{2} r^2 \theta_{\text{rad}} = \pi r^2 \times \left(\frac{\theta^\circ}{360^\circ}\right)$.
• Chord Length ($c$): $c = 2r \sin\left(\frac{\theta}{2}\right)$.
• Circular Segment Area: $A_{\text{segment}} = A_{\text{sector}} - A_{\text{triangle}} = \frac{1}{2} r^2 (\theta_{\text{rad}} - \sin(\theta_{\text{rad}}))$.

Arc Length vs Sector vs Segment