returns a filled triangle with sides of length a and b and angle γ between them.

Details and Options

  • SASTriangle is also known as side-angle-side triangle.
  • SASTriangle can be used as a primitive in 2D graphics and as a geometric region in 2D.
  • The given (blue) and computed (red) parameters for an SASTriangle:
  • SASTriangle returns a Triangle with A at the origin, B on the positive axis, and C in the half-plane .
  • SASTriangle allows the lengths a and b to be any positive numbers and the angle γ strictly between 0 and .

Background & Context

  • SASTriangle constructs a side-angle-side triangle. In particular, SASTriangle[a,γ,b] represents the Triangle in TemplateBox[{}, Reals]^2 with vertices , and located at the origin, on the positive axis and in the upper half-plane, respectively, with a and b the lengths of the sides opposite vertices and and γ∠ACB. By the SAS theorem, the triangle so specified is unique (up to geometric congruence). SASTriangle allows the lengths a and b to be any positive numbers and the angle γ to be a positive value satisfying . The arguments of SASTriangle may be exact or approximate numeric expressions.
  • The Triangle objects returned by SASTriangle can be used as 2D graphics primitives or geometric regions.
  • SASTriangle is related to a number of other symbols. AASTriangle, ASATriangle and SSSTriangle return two-dimensional triangles constructed using different angle and/or side specifications. SASTriangle is a special case of Triangle, in the sense that SASTriangle[a,γ,b] is equivalent to Triangle[{{0,0},{x,0},{y,z}}] for xSqrt[a^2+b^2-2 a b Cos[γ]], y(b^2-a bCos[γ])/Sqrt[a^2+b^2-2 a b Cos[γ]] and z(a b Sin[γ])/Sqrt[a^2+b^2-2 a b Cos[γ]].


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Basic Examples  (4)

A triangle with , , and :

An SASTriangle:

Different styles applied to an SASTriangle:

Area and centroid:

Scope  (14)

Graphics  (4)

Specification  (2)

SASTriangle evaluates to Triangle with one point at the origin and one edge on the axis:

A triangle with symbolic edge length:

Plot them:

Styling  (2)

Color directives specify the face color:

FaceForm and EdgeForm can be used to specify the styles of the interior and boundary:

Regions  (10)

Embedding dimension is the dimension of the space the triangle lives in:

Geometric dimension is the dimension of the shape itself:

Membership testing:

Conditions for membership:



Distance from a point to an SASTriangle:

Visualize it:

Signed distance from a point:

Now plot it:

Nearest point:

Visualize it:

A triangle is bounded:

Find its range:

Integrate over an SASTriangle:

Optimize over it:

Solve equations over an SASTriangle:

Applications  (2)

A triangle with two equal sides is an isosceles triangle:

Visualize it:

Find its area:

The circumcircle of an SASTriangle can be found using Circumsphere:

The circumcircle passes through the three corner points:

Find the midpoints for each edge of the triangle:

The perpendicular bisectors are lines from the circumcenter to the midpoints:

Properties & Relations  (2)

SASTriangle is a specialized case of Triangle:

Any SASTriangle can be represented by a Polygon:

Neat Examples  (1)

Varying one angle:

Wolfram Research (2014), SASTriangle, Wolfram Language function,


Wolfram Research (2014), SASTriangle, Wolfram Language function,


Wolfram Language. 2014. "SASTriangle." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (2014). SASTriangle. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2024_sastriangle, author="Wolfram Research", title="{SASTriangle}", year="2014", howpublished="\url{}", note=[Accessed: 17-June-2024 ]}


@online{reference.wolfram_2024_sastriangle, organization={Wolfram Research}, title={SASTriangle}, year={2014}, url={}, note=[Accessed: 17-June-2024 ]}