# Angle Command

##### Command Categories (All commands)

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 \mathrm{\mathsf{ 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 \mathrm{\mathsf{ 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.