Elliptical Sector Area Calculator

Enter the semi-major and semi-minor axes plus the start and end angles from the center to find the area of an elliptical sector, with the full ellipse area and swept fraction.

Input

Enter the ellipse semi-major and semi-minor axes plus the start and end angles in degrees from the center to find the elliptical sector area.

Semi-major axis a

Semi-minor axis b

Start angle (deg)

End angle (deg)

Result

Sector area

23.258619

Full ellipse area

75.398224

Angular sweep

120 deg

Fraction covered

30.847701%

Length units match the inputs and the area is in those units squared. Angles are positive counterclockwise.

How it works

A point on the ellipse uses the parametric angle psi as x=a cos psi and y=b sin psi, where a is the semi-major axis and b the semi-minor axis.
The polar angle phi and the parametric angle psi are linked by tan phi=(b sin psi)/(a cos psi), so psi=atan2(a sin phi, b cos phi).
The elliptical sector area uses the difference of parametric angles: area=(ab/2)(psi2 minus psi1).
The full ellipse area is pi a b. The fraction covered by the sector equals area divided by pi a b.
Angles use standard position with zero pointing right and counterclockwise positive. When the end angle is at or below the start, a full turn is added to the sweep.
Length units match the inputs, and the area is in those units squared.

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