Cirkelberegner

Sådan bruges denne lommeregner

1. Indtast Radius (r)
2. Klik på knappen Beregn
3. Læs resultatet vist under lommeregneren

Circle Formulas: Area, Circumference, and Diameter

A circle is the set of all points equidistant from a center point, with that distance called the radius (r). The diameter (d) is twice the radius: d = 2r. The three main measurements of a circle are related through the mathematical constant π (pi) ≈ 3.14159265.

Area = πr² — the space enclosed within the circle. Circumference = 2πr = πd — the perimeter or distance around the circle. Diameter = 2r — the longest chord through the center. If you know any one measurement, you can find all others through these formulas.

π is an irrational number — its decimal expansion never repeats or terminates: 3.14159265358979... For most engineering calculations, using π ≈ 3.14159 (5 decimal places) gives results accurate to 5 significant figures. For precise scientific work, use more digits. Our calculator uses JavaScript's Math.PI = 3.141592653589793.

Sectors, Arcs, and Partial Circles

A sector is a 'pie slice' of a circle defined by a central angle θ. Sector area = (θ/360°) × πr² for θ in degrees, or (θ/2) × r² for θ in radians. An arc is the curved boundary of a sector: arc length = (θ/360°) × 2πr.

A segment is the region between a chord and its arc. A chord is any line segment with both endpoints on the circle. The longest chord is the diameter. The relationship between a chord's length (c) and its distance from the center (d): c = 2√(r² - d²).

Radians are the natural angle unit for circles: 2π radians = 360°. One radian is the angle subtended by an arc equal in length to the radius. Radians simplify formulas: arc length = rθ, sector area = r²θ/2. Calculus and trigonometry almost always use radians because derivatives of sin and cos are simple only in radians.

Circles in Real Life and Advanced Mathematics

Circles are ubiquitous in engineering, science, and everyday life. Wheels, gears, pipes, coins, and planetary orbits are approximately circular. The circle maximizes area for a given perimeter (isoperimetric inequality) — this is why bubbles are round and why cylindrical containers are efficient.

The equation of a circle with center (h,k) and radius r is (x-h)² + (y-k)² = r². In trigonometry, the unit circle (r=1, center at origin) defines sine and cosine: for an angle θ, the point on the unit circle is (cos θ, sin θ). This makes the unit circle the foundation of all of trigonometry.

In higher mathematics, circles are special cases of conic sections (the intersection of a cone with a plane perpendicular to its axis). Non-Euclidean geometry generalizes the circle concept to curved spaces. In complex analysis, Möbius transformations map circles to circles (or lines, which are circles through infinity), making circles fundamental to complex geometry.