PlanarAngle

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".

Details

  • 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.

Examples

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Basic Examples  (2)

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

The angle formed by a triangle at origin:

Scope  (7)

Basic Uses  (2)

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

PlanarAngle works with numeric arguments:

Symbolic arguments:

Specifications  (5)

"Counterclockwise"  (1)

The angle formed by a counterclockwise rotation:

"Clockwise"  (1)

The angle formed by a clockwise rotation:

"Interior"  (1)

The interior angle of a triangle at the origin:

"Exterior"  (1)

The exterior angle of a triangle at the origin:

"FullExterior"  (1)

The full exterior angle of a triangle at the origin:

Applications  (6)

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:

Properties & Relations  (7)

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}:

Possible Issues  (1)

PlanarAngle gives generic values for symbolic parameters:

Wolfram Research (2019), PlanarAngle, Wolfram Language function, https://reference.wolfram.com/language/ref/PlanarAngle.html.

Text

Wolfram Research (2019), PlanarAngle, Wolfram Language function, https://reference.wolfram.com/language/ref/PlanarAngle.html.

CMS

Wolfram Language. 2019. "PlanarAngle." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/PlanarAngle.html.

APA

Wolfram Language. (2019). PlanarAngle. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PlanarAngle.html

BibTeX

@misc{reference.wolfram_2025_planarangle, author="Wolfram Research", title="{PlanarAngle}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/PlanarAngle.html}", note=[Accessed: 29-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_planarangle, organization={Wolfram Research}, title={PlanarAngle}, year={2019}, url={https://reference.wolfram.com/language/ref/PlanarAngle.html}, note=[Accessed: 29-May-2025 ]}