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PlanarAngle
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PlanarAngle

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PlanarAngle[p{q1,q2}]

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

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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

open allclose all

Basic Examples  (2)Summary of the most common use cases

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

In[1]:=1
✖
https://wolfram.com/xid/0n4mw4d17jq-5qtb0u
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0n4mw4d17jq-5mgpw7
Out[2]=2

The angle formed by a triangle at origin:

In[1]:=1
✖
https://wolfram.com/xid/0n4mw4d17jq-25x74j
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0n4mw4d17jq-tikovj
Out[2]=2

Scope  (7)Survey of the scope of standard use cases

Basic Uses  (2)

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

In[1]:=1
✖
https://wolfram.com/xid/0n4mw4d17jq-n05why
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0n4mw4d17jq-x01uu4
Out[2]=2

PlanarAngle works with numeric arguments:

In[1]:=1
✖
https://wolfram.com/xid/0n4mw4d17jq-lvqgwi
Out[1]=1

Symbolic arguments:

In[2]:=2
✖
https://wolfram.com/xid/0n4mw4d17jq-ghrjq7
Out[2]=2

Specifications  (5)

"Counterclockwise"  (1)

The angle formed by a counterclockwise rotation:

In[1]:=1
✖
https://wolfram.com/xid/0n4mw4d17jq-nv7vl0
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0n4mw4d17jq-lhkues
Out[2]=2

"Clockwise"  (1)

The angle formed by a clockwise rotation:

In[3]:=3
✖
https://wolfram.com/xid/0n4mw4d17jq-nsa0zr
Out[3]=3
In[2]:=2
✖
https://wolfram.com/xid/0n4mw4d17jq-cuvk4h
Out[2]=2

"Interior"  (1)

The interior angle of a triangle at the origin:

In[1]:=1
✖
https://wolfram.com/xid/0n4mw4d17jq-qre3u9
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0n4mw4d17jq-qpf6rl
Out[2]=2

"Exterior"  (1)

The exterior angle of a triangle at the origin:

In[1]:=1
✖
https://wolfram.com/xid/0n4mw4d17jq-0hh5ns
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0n4mw4d17jq-4nq4jx
Out[2]=2

"FullExterior"  (1)

The full exterior angle of a triangle at the origin:

In[1]:=1
✖
https://wolfram.com/xid/0n4mw4d17jq-d3dhl6
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0n4mw4d17jq-cqqzvo
Out[2]=2

Applications  (6)Sample problems that can be solved with this function

A straight angle:

In[1]:=1
✖
https://wolfram.com/xid/0n4mw4d17jq-9fbw0g
In[2]:=2
✖
https://wolfram.com/xid/0n4mw4d17jq-n4ifn6
Out[2]=2

It is an angle of π:

In[3]:=3
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https://wolfram.com/xid/0n4mw4d17jq-vj7szl
Out[3]=3

An obtuse angle:

In[1]:=1
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https://wolfram.com/xid/0n4mw4d17jq-matwcn
In[2]:=2
✖
https://wolfram.com/xid/0n4mw4d17jq-694z52
Out[2]=2

It is an angle between and π:

In[3]:=3
✖
https://wolfram.com/xid/0n4mw4d17jq-bphixo
Out[3]=3

A right angle:

In[1]:=1
✖
https://wolfram.com/xid/0n4mw4d17jq-uobo2d
In[2]:=2
✖
https://wolfram.com/xid/0n4mw4d17jq-f0w0vc
Out[2]=2

It is an angle of :

In[3]:=3
✖
https://wolfram.com/xid/0n4mw4d17jq-vxs71l
Out[3]=3

An acute angle:

In[1]:=1
✖
https://wolfram.com/xid/0n4mw4d17jq-e0wgir
In[2]:=2
✖
https://wolfram.com/xid/0n4mw4d17jq-2ktu1n
Out[2]=2

It is an angle smaller than :

In[3]:=3
✖
https://wolfram.com/xid/0n4mw4d17jq-sryusk
Out[3]=3

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

In[1]:=1
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https://wolfram.com/xid/0n4mw4d17jq-odig9z
In[2]:=2
✖
https://wolfram.com/xid/0n4mw4d17jq-cxyjin
In[3]:=3
✖
https://wolfram.com/xid/0n4mw4d17jq-hjo7x3
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0n4mw4d17jq-7jf61k
Out[4]=4

An AASTriangle:

In[1]:=1
✖
https://wolfram.com/xid/0n4mw4d17jq-nufs4f
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0n4mw4d17jq-ugmnb
Out[2]=2

Get the angles:

In[3]:=3
✖
https://wolfram.com/xid/0n4mw4d17jq-ztmib3
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0n4mw4d17jq-8l3cjo
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0n4mw4d17jq-uc5opr
Out[5]=5

Properties & Relations  (7)Properties of the function, and connections to other functions

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

In[1]:=1
✖
https://wolfram.com/xid/0n4mw4d17jq-414zxc
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0n4mw4d17jq-0xtjvx
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0n4mw4d17jq-gav70s
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0n4mw4d17jq-xwkpin
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0n4mw4d17jq-knnf7m
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0n4mw4d17jq-pac8dj
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0n4mw4d17jq-p3qmdf
In[2]:=2
✖
https://wolfram.com/xid/0n4mw4d17jq-jg3w5h
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0n4mw4d17jq-qqzulz
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0n4mw4d17jq-fsm0rj
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0n4mw4d17jq-h3aavt
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0n4mw4d17jq-80zo14
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0n4mw4d17jq-1kxdnt
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0n4mw4d17jq-0uz03d
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0n4mw4d17jq-chwid9
Out[3]=3

Possible Issues  (1)Common pitfalls and unexpected behavior

PlanarAngle gives generic values for symbolic parameters:

In[1]:=1
✖
https://wolfram.com/xid/0n4mw4d17jq-ykpc2
Out[1]=1
Wolfram Research (2019), PlanarAngle, Wolfram Language function, https://reference.wolfram.com/language/ref/PlanarAngle.html.
✖
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.

✖
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.

✖
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

✖
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: 05-May-2025 ]}

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@misc{reference.wolfram_2025_planarangle, author="Wolfram Research", title="{PlanarAngle}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/PlanarAngle.html}", note=[Accessed: 05-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: 05-May-2025 ]}

✖
@online{reference.wolfram_2025_planarangle, organization={Wolfram Research}, title={PlanarAngle}, year={2019}, url={https://reference.wolfram.com/language/ref/PlanarAngle.html}, note=[Accessed: 05-May-2025 ]}