Circle Theorems Calculator

How Circle Theorems Calculator Works?

The Circle Theorems Calculator uses the radius r and central angle θ to calculate fundamental circle measurements. The radius must be positive, and the central angle can be entered in degrees or radians. Internally, angle-based formulas use radians, so a degree angle is represented mathematically by $\theta =\alpha \frac{\pi }{180}$, where α is the angle in degrees.

The diameter is twice the radius. The circumference is the complete distance around the circle.

For a central angle θ, the corresponding arc length is the fraction of the circumference represented by that angle. The chord is the straight-line segment joining the arc endpoints.

The inscribed-angle theorem states that an angle at the circumference subtending an arc is half the corresponding central angle. The tangent-chord angle has the same magnitude by the alternate segment theorem.

The sector area is the fraction of the full circle selected by the central angle. The minor segment is the sector minus the isosceles triangle formed by the two radii and the chord.

The sagitta is the height of the arc above the chord, while the center-to-chord distance is the perpendicular distance from the center to the chord.

Practical Use Cases

Students can enter a radius and central angle to check solutions for chord length, arc length, sector area, and circle theorem exercises involving inscribed or tangent-chord angles.

Mathematics teachers can use the calculator to create examples that compare central angles with their corresponding inscribed angles and demonstrate how changing the radius affects lengths and areas.

Engineers and designers can estimate curved paths, circular cutouts, sector-shaped components, chord spans, and the height of an arc above a chord using the sagitta result.

Surveyors and technical planners can examine circular boundaries and partial arcs by calculating the distance around a selected arc, the straight chord between endpoints, and the perpendicular center-to-chord distance.

Geometry learners can explore the relationship between a sector and its minor segment, seeing how the triangle formed by two radii is subtracted from the sector area.

Accuracy and Limitations

The calculator uses standard trigonometric circle formulas and the mathematical constant π, so results are highly accurate within normal floating-point precision and are rounded for display. The radius must be between 0.0001 and 1,000,000,000, while the central angle must be between 0.01 and 360 degrees or its radian equivalent. Angle-based calculations assume a single central angle describing the selected arc; the minor-segment formula is geometrically applicable to a minor central angle up to 180 degrees. For an angle greater than 180 degrees, interpret the corresponding result as belonging to the selected larger arc or use its supplementary minor angle when a true minor segment is required. The calculator does not account for measurement uncertainty, drawing scale, material thickness, or non-circular geometry.

Quick Examples