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PlanarAngle—Wolfram Language Documentation

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• See Also
  • VectorAngle
  • SolidAngle
  • DihedralAngle
  • PolygonAngle
  • PolyhedronAngle
  • Dot
  • AnglePath
  • AngleBisector
  • GeometricScene
• Related Guides
  • Region Properties and Measures
  • Synthetic Geometry
  • Angles and Polar Coordinates
  • Plane Geometry

PlanarAngle[p{q₁,q₂}]

gives the angle between the half‐lines from p through q₁ and q₂.

PlanarAngle[{q₁,p,q₂}]

gives the angle at p formed by the triangle with vertex points p, q₁ and q₂.

PlanarAngle[…,"spec"]

gives the angle specified by "spec".

• PlanarAngle is also known as angle.
• PlanarAngle[p{q₁,q₂}] gives the length of the arc of the unit circle Circle[p] delimited by the half-line from p through q₁ on the left and the half-line from p to q₂ on the right.
• Two half‐lines from p through q₁ and q₂ delimit two angles α₁ and α₂ at p.
• The following specifications "spec" can be given:
• "Counterclockwise" angle formed by the counterclockwise rotation from q₁ to q₂ "Clockwise" angle formed by the clockwise rotation from q₁ to q₂
• PlanarAngle[p{q₁,q₂},"Counterclockwise"] is equivalent to PlanarAngle[p{q₁,q₂}].
• PlanarAngle[p{q₁,q₂},"Clockwise"] is equivalent to PlanarAngle[p{q₂,q₁}].
• PlanarAngle[{q₁,p,q₂}] is the angle subtended by the line segment q₁ q₂ from p.
• The triangle with vertex points q₁, p and q₂ defines three angles α₁, α₂ and α₃ at p.
• The following specifications "spec" can be given:
• "Interior" interior (inside) angle of the triangle at p "Exterior" exterior angle of the triangle at p "FullExterior" full exterior angle of the triangle at p
• PlanarAngle[{q₁,p,q₂},"Interior"] is equivalent to PlanarAngle[{q₁,p,q₂}].
• PlanarAngle[{q₁,p,q₂},"Exterior"] is equivalent to π-PlanarAngle[{q₁,p,q₂}].
• PlanarAngle[{q₁,p,q₂},"FullExterior"] is equivalent to 2π-PlanarAngle[{q₁,p,q₂}].
• With the specification "Interior", "Exterior" or "FullExterior", PlanarAngle[p{q₁,q₂},"spec"] is taken to be PlanarAngle[{q₁,p,q₂},"spec"].
• With the specification "Counterclockwise" or "Clockwise", PlanarAngle[{q₁,p,q₂},"spec"] is taken to be PlanarAngle[p{q₁,q₂}, "spec"].
• PlanarAngle can be used with symbolic points in GeometricScene.

The angle between the half‐lines from {0,0} through {1,1} and {1,0}:

The angle formed by a triangle at origin:

Use PlanarAngle to find the angle between two half‐lines:

PlanarAngle works with numeric arguments:

Symbolic arguments:

The angle formed by a counterclockwise rotation:

The angle formed by a clockwise rotation:

The interior angle of a triangle at the origin:

The exterior angle of a triangle at the origin:

The full exterior angle of a triangle at the origin:

A straight angle:

It is an angle of π:

An obtuse angle:

It is an angle between and π:

A right angle:

It is an angle of :

An acute angle:

It is an angle smaller than :

Find the interior angle of a triangle at a point p:

An AASTriangle:

Get the angles:

PlanarAngle[p,{q₂,q₁}] is equal to 2π-PlanarAngle[p,{q₁,q₂}]:

PlanarAngle[{q₁,p,q₂},"Interior"] is the smallest angle formed by the rotations around p:

PlanarAngle[p{q₁,q₂}] takes values from 0 to 2π:

PlanarAngle[{q₁,p,q₂}] takes values from 0 to π:

Dihedral angle is the planar angle in the plane defined by the normal p₂-p₁ and a point p₁:

PlanarAngle[p->{q₁,q₂}] is equivalent to PolygonAngle[ℛ, p] where q₁ and q₂ are adjacent points of p in a polygon ℛ:

PlanarAngle[{q₁,p,q₂}] is equivalent to SolidAngle[p,{q₁,q₂}:

PlanarAngle gives generic values for symbolic parameters:

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