Calculate circle area, circumference, diameter, radius, arc length, and sector area with step-by-step geometric derivations.
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A circle is the set of all points in a two-dimensional plane that are equidistant from a central point. The distance from the center to any point on the boundary is the radius (r), and the boundary distance is the circumference (C).
A = \pi r^2, \quad C = 2\pi r = \pi d, \quad A_{\text{sector}} = \frac{\theta}{360^\circ}\pi r^2Area is calculated by multiplying pi by radius squared. Circumference is 2 times pi times radius. Sector area is the proportion of the central angle theta over 360 degrees.
Scenario: Find area and circumference of a circle with radius r = 7 cm
Diameter: d = 2 × 7 = 14 cm
Circumference: C = 2 × π × 7 ≈ 43.9823 cm
Area: A = π × (7)² = 49π ≈ 153.938 cm²
The diameter is exactly twice the length of the radius: d = 2r, or r = d / 2.
Multiply the full circle area (πr²) by the fraction of the circle subtended by the central angle: (θ / 360°).
Arc length is the curved distance along the circumference corresponding to a central angle: s = (θ / 360°) × 2πr.
Pi (π ≈ 3.14159) represents the universal ratio of any circle’s circumference to its diameter (C / d).
Calculate sphere volume, surface area, diameter, and great circle circumference from radius with step-by-step mathematical deduction.
Calculate right circular cylinder volume, total surface area, lateral (curved) surface area, and base area from radius and height.
Calculate triangle area using Heron’s formula or base and height, perimeter, semi-perimeter, inradius, and classification.