# PlanarAngle

PlanarAngle[p{q1,q2}]

gives the angle between the halflines 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 halflines 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)

The angle between the halflines 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 halflines:

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[, p] where q1 and q2 are adjacent points of p in a polygon :

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

## Possible Issues(1)

PlanarAngle gives generic values for symbolic parameters: