TRI_AAAS

This function solves a triangle when all three angles and one side length are known. The known side sets the scale, and the remaining sides follow from the Law of Sines.

With known angles and one side f opposite angle F:

\frac{d}{\sin(D)} = \frac{e}{\sin(E)} = \frac{f}{\sin(F)}

This determines side lengths uniquely when the angle set is valid for a triangle.

Excel Usage

=TRI_AAAS(D, E, F, f)

D (float, required): First known angle (radians).
E (float, required): Second known angle (radians).
F (float, required): Third known angle (radians).
f (float, required): Known side opposite angle F (length units).

Returns (list[list]): A 2D array where the first row is [d, e, f] and the second row is [D, E, F].

Example 1: Solve from angle triplet summing to pi with side scale one

Inputs:

D	E	F	f
1	1.1	1.041592653589793	1

Excel formula:

=TRI_AAAS(1, 1.1, 1.041592653589793, 1)

Expected output:

Result
0.974817	1.03243	1
1	1.1	1.04159

Example 2: Solve from balanced angle set and larger known side

Inputs:

D	E	F	f
0.9	1	1.2415926535897932	12

Excel formula:

=TRI_AAAS(0.9, 1, 1.2415926535897932, 12)

Expected output:

Result
9.93334	10.6707	12
0.9	1	1.24159

Example 3: Solve from mixed-angle geometry with known opposite side

Inputs:

D	E	F	f
0.5	1.2	1.4415926535897932	7

Excel formula:

=TRI_AAAS(0.5, 1.2, 1.4415926535897932, 7)

Expected output:

Result
3.38419	6.57911	7
0.5	1.2	1.44159

Example 4: Solve from equal first two angles and scaled side

Inputs:

D	E	F	f
1	1	1.1415926535897931	9

Excel formula:

=TRI_AAAS(1, 1, 1.1415926535897931, 9)

Expected output:

Result
8.32867	8.32867	9
1	1	1.14159

Python Code

external_packages = ['trianglesolver==1.2']

from trianglesolver import aaas as trianglesolver_aaas

def tri_aaas(D, E, F, f):
    """
    Solve a triangle from three angles and one side for scale.

    See: https://pypi.org/project/trianglesolver/

    This example function is provided as-is without any representation of accuracy.

    Args:
        D (float): First known angle (radians).
        E (float): Second known angle (radians).
        F (float): Third known angle (radians).
        f (float): Known side opposite angle F (length units).

    Returns:
        list[list]: A 2D array where the first row is [d, e, f] and the second row is [D, E, F].
    """
    try:
        result = trianglesolver_aaas(D, E, F, f)
        return [[result[0], result[1], result[2]], [result[3], result[4], result[5]]]
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

D *

First known angle (radians).

E *

Second known angle (radians).

F *

Third known angle (radians).

f *

Known side opposite angle F (length units).