Angle Command

Angle( <Object> )

• Conic: Returns the angle of twist of a conic section’s major axis (see command Axes).

Example: Angle(x²/4+y²/9=1) yields 90° or 1.57 if the default angle unit is radians.

Note: It is not possible to change the Angle Unit to Radian in GeoGebra 5.0 Web and Tablet App Version.

• Vector: Returns the angle between the x‐axis and given vector.

Example: Angle(Vector((1, 1))) yields 45° or the corresponding value in radians.

• Point: Returns the angle between the x‐axis and the position vector of the given point.

Example: Angle((1, 1)) yields 45° or the corresponding value in radians.

• Number: Converts the number into an angle (result in [0,360°] or [0,2π] depending on the default angle unit).

Example: Angle(20) yields 65.92° when the default unit for angles is degrees.

• Polygon: Creates all angles of a polygon in mathematically positive orientation (counter clockwise).

Example: Angle(Polygon((4, 1), (2, 4), (1, 1))) yields 56.31°, 52.13° and 71.57° or the corresponding values in radians.

Note: If the polygon was created in counter clockwise orientation, you get the interior angles. If the polygon was created in clockwise orientation, you get the exterior angles.

Angle( <Vector>, <Vector> )

Returns the angle between two vectors (result in [0,360°] or [0,2π] depending on the default angle unit).

Example:

Angle(Vector((1, 1)), Vector((2, 5))) yields 23.2° or the corresponding value in radians.

Angle( <Line>, <Line> )

Returns the angle between the direction vectors of two lines (result in [0,360°] or [0,2π] depending on the default angle unit).

Example:

• Angle(y = x + 2, y = 2x + 3) yields 18.43° or the corresponding value in radians..
• Angle(Line((-2, 0, 0), (0, 0, 2)), Line((2, 0, 0), (0, 0, 2))) yields 90° or the corresponding value in radians.

and in CAS View :

• Angle(x + 2, 2x + 3) yields acos \left( 3 \cdot \frac{\sqrt{10}}{10} \right).
• Define f(x) := x + 2 and g(x) := 2x + 3 then command Angle(f(x), g(x)) yields acos \left(3 \cdot \frac{\sqrt{10}}{10} \right).

Angle( <Line>, <Plane> )

Returns the angle between the line and the plane.

Example:

• Angle(Line((1, 2, 3),(-2, -2, 0)), z = 0) yields 30.96° or the corresponding value in radians.

Angle( <Plane>, <Plane> )

Returns the angle between the two given planes.

Example:

• Angle(2x - y + z = 0, z = 0) yields 114.09° or the corresponding value in radians.

Angle( <Point>, <Apex>, <Point> )

Returns the angle defined by the given points (result in [0,360°] or [0,2π] depending on the default angle unit).

Example:

Angle((1, 1), (1, 4), (4, 2)) yields 56.31° or the corresponding value in radians.

Angle( <Point>, <Apex>, <Angle> )

Returns the angle of size α drawn from point with apex.

Example:

:*Angle((0, 0), (3, 3), 30°) yields 30° and the point (1.9, -1.1).

Note: The point Rotate( <Point>, <Angle>, <Apex> ) is created as well.

Angle( <Point>, <Point>, <Point>, <Direction> )

Returns the angle defined by the points and the given Direction, that may be a line or a plane (result in [0,360°] or [0,2π] depending on the default angle unit).

Note: Using a Direction allows to bypass the standard display of angles in 3D which can be set as just [0,180°] or [180°,360°], so that given three points A, B, C in 3D the commands Angle(A, B, C) and Angle(C, B, A) return their real measure instead of the one restricted to the set intervals.

Example:

Angle((1, -1, 0),(0, 0, 0),(-1, -1, 0), zAxis) yields 270° and Angle((-1, -1, 0),(0, 0, 0),(1, -1, 0), zAxis) yields 90° or the corresponding values in radians.

Note: See also Angle and Angle with Given Size tools.