What is an AAS triangle?

AAS (Angle-Angle-Side) refers to a triangle in which two angles and one side are known, with the known side being opposite to one of the known angles. Since the third angle is determined by the two known angles (C = 180 - A - B), and the overall size of the triangle is fixed by the known side, an AAS triangle is always uniquely defined. This means that there is exactly one triangle that satisfies these conditions.

How to solve an AAS triangle?

First calculate the third angle by subtracting the sum of the two known angles from 180 degrees. Then use the law of sines to determine a common factor based on the known sides and their opposite angles. This common factor is then multiplied by the sine values of the other angles to find the lengths of the remaining sides. Finally, calculate the area, perimeter, and altitudes.

What is the difference between AAS and ASA?

Both have the condition of two angles and one side. In AAS, the known side is opposite to one of the known angles, so it's a non-included side. In ASA, the known side is between the two angles, so it's an included side. The solution methods are the same, and since both give exactly one triangle, they can be worked with this tool. You just have to choose which side is known.

What is the area of an AAS triangle with a 40-degree angle, a 25-degree angle and a side length of 16 centimeters?

For a triangle with angle A = 40 degrees, angle B = 25 degrees and side a = 16 cm, the third angle is C = 115 degrees. The formula for area is Area = 1/2*a squared*Sinus(B) * Sinus(A+B) / Sinus(A)" results in an area of approximately 76.27 square centimeters. The remaining two sides are each approximately b = 10.52 cm and c = 22.56 cm long.

How to calculate the height of an AAS triangle?

First calculate the missing angle C = 180 - A - B. The height to a side is twice the area divided by that side. With an equivalent formula this can be calculated in one step: the height to side b is a * Sin(C). Before plugging values in, make sure all angles are in the same units, either degrees or radians.

Can any AAS triangle be solved?

An AAS triangle can only be solved if both input angles are greater than 0 degrees and their sum is less than 180 degrees. This is because the third angle must be positive. If the sum of the two angles is greater than or equal to 180 degrees, then no triangle exists, and the tool will output a corresponding message instead of an invalid result. For any valid combination of angles with positive lengths, there is exactly one triangle that can be formed.

Can this tool be used to determine what type of triangle it is?

Yes it is possible. Based on the angles a tool can classify a triangle as acute triangle(all angles are less than 90 degrees), right triangle(one angle is 90 degrees) or obtuse triangle(one angle is greater than 90 degrees). Also based on the length of sides a tool can determine if a triangle is scalene triangle, isosceles triangle (two angles are equal) or equilateral triangle(all angles are 60 degrees).

AAS Triangle Calculator

Your known values

Enter two angles and the one side you know. Angle A is opposite side a, B opposite b, C opposite c - keep those pairs straight and the law of sines does the rest.

Which side do you know?

AAS means the known side sits opposite one of the two known angles. If it sits between them instead (the ASA case), pick side c - the maths is the same and the answer is still unique.

Side a - opposite angle A (AAS)

Side b - opposite angle B (AAS)

Side c - between A and B (ASA)

Angle A

Angle B

Known side length

Solved triangle

Angle C (third angle)

Side a

Side b

Side c

Area

Perimeter

Show heights and circle radii

The perpendicular height to each side, the circumscribed and inscribed circle radii, and the semi-perimeter.

This is an obtuse triangle - one angle is greater than 90°.

Area

m²

Third angle C

deg (°)

Solved triangle: angles A = 40 deg (°), B = 25 deg (°), C = 115 deg (°); sides a = 16 m, b = 10.52 m, c = 22.56 m. Area 76.27, perimeter 49.08.

Steps and chart

Show the calculation steps

Walk through how two angles and one side unlock the whole triangle.

Show the side-length bar chart

Compare the three side lengths of the solved triangle at a glance.

AAS is an abbreviation of Angle-Angle-Side and means that two angles and one side of a triangle are known. If these conditions are met then exactly one triangle can be determined unambiguously. This tool allows you to simply enter three values, in order to calculate the length of all sides, angles, height, area and perimeter.

The sum of the three interior angles of a triangle is always 180 degrees, and this is key to solving. If two angles are known, then the third angle can be determined. Once all three angles are established, the size of the triangle will be determined by the known side, and the remaining two sides can be calculated proportionally using the law of sines.

The importance of AAS and the difference to ASA:

Both AAS and ASA have two angles and one side known, but the difference is in the position of the known side. In AAS, the known side is opposite to one of the known angles, so it's a non-included side. In ASA, the known side is between the two known angles, so it's an included side.

You know	Case	Where the side sits	Solutions
Two angles + a non-included side	AAS	Opposite a known angle	Exactly 1
Two angles + the included side	ASA	Between the two angles	Exactly 1

The solution methods for both cases are the same, and since only one triangle is possible, this tool can be used. You just need to specify which side is known in the top drop-down field. AAS is also one of the congruence conditions for triangles. Any two triangles with the same AAS conditions are completely identical in size and shape.

The calculation formula for AAS is as follows:

All calculations are based on the law of sines. This states that the ratio of any side to the sine of its opposite angle is the same for all three sides.

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R

This common ratio is not just an abstract number; it corresponds to 2R and represents twice the radius of the circumcircle of the triangle. Solving for AAS triangles follows a short, fixed procedure.

C = 180^\circ - A - B

First find one known pair of corresponding sides and angles, then create the ratio of the side length to the sine of its opposite angle, and then use that ratio to solve for another side. For example, if side a and its opposite angle A are known, the following formula applies:

b = \frac{a\,\sin B}{\sin A}, \qquad c = \frac{a\,\sin C}{\sin A}

Example of a calculation:

We are solving for an AAS triangle with side a=16 cm, angle A=40 degrees and angle B=25 degrees. First we calculate the third angle: C = 180 - 40 - 25 = 115 degrees. Next we use the known side a and angle A as reference to solve for the remaining two sides using proportions.

b = \frac{16\,\sin 25^\circ}{\sin 40^\circ} \approx 10.52\ \text{cm}, \qquad c = \frac{16\,\sin 115^\circ}{\sin 40^\circ} \approx 22.56\ \text{cm}

This gives us an area of approximately 76.27 square centimeters and a perimeter of approximately 49.08 centimeters. If you enter the values A = 40, B = 25, and side a = 16 into the input fields, the tool will return these values and also show the height as well as the radii of the inscribed and circumscribed circles for each side.

How to calculate area of AAS triangle:

If two sides and the included angle are known, then the area can be directly calculated. There is also a simple closed-form solution to compute the area of an AAS triangle where only the known side a and two angles are used.

\text{Area} = \tfrac{1}{2}\,a^2\,\frac{\sin B\,\sin(A+B)}{\sin A}

In the example above, this would be calculated as follows: Take half of the square of 16, multiply it by the sine values of 25 degrees and 65 degrees, then divide that result by the sine value of 40 degrees. The resulting area is again about 76.27 cm².

The height of an AAS triangle:

Any side of a triangle has an associated altitude. This is the perpendicular distance from the opposite vertex to the line containing that side. The quickest way to calculate it is by using the area directly. An equivalent formula can also be created using a known side and the sine value of one of the angles.

h_a = \frac{2\,\text{Area}}{a}, \quad h_b = \frac{2\,\text{Area}}{b}, \quad h_c = \frac{2\,\text{Area}}{c} \qquad\text{e.g. } h_b = a\,\sin C

The tool shows three heights, and you can select the appropriate result based on which side of the base is desired.

Step-by-step solution

Any issue on AAS follows the same short procedure. The tool can also show the full process below the answer.

1. Add the two known angles and subtract that sum from 180 degrees to find the third angle. 2. Assign the known side to its opposite angle and form a ratio of the length of this side to the sine of this angle. 3. Multiply this ratio by the sines of each remaining angle to determine the other two sides. 4. Calculate the area from the two sides and the included angle, then the perimeter, any altitudes, and the radii of the inscribed and circumscribed circles. 5. Check your work: The longest side must be opposite the largest angle, and the sum of the three interior angles must equal 180 degrees.

Triangle Classification

Since AAS begins with conditions on the angles, it can also determine what type of triangle is being dealt with before determining the ratio of side lengths. In terms of classifying by angles, if all three angles are less than 90 degrees then it is an acute triangle; if one angle is exactly 90 degrees then it is a right triangle; and if one angle is greater than 90 degrees then it is an obtuse triangle.

For the side classification, if all three sides are different lengths, it is a scalene triangle; if two sides are equal in length, it is an isosceles triangle; and if all three sides are equal in length, it is an equilateral triangle. With AAS conditions, two congruent angles always correspond to two congruent sides, leading to an isosceles triangle. Three 60-degree angles will always result in an equilateral triangle.

Area, circumference and two circles

For any triangle that has a solution, there are two circles associated with it. The circumradius R is the radius of the circle that passes through all three vertices of the triangle and can be found by the formula R = a / (2 sin A) where a is one side of the triangle and A is the angle opposite that side. The inradius r is the radius of the largest circle that fits inside the triangle, and can be calculated from its area and semiperimeter using the formula r = K / s where K is the area of the triangle and s is half its perimeter.

R = \frac{a}{2\sin A}, \qquad r = \frac{\text{Area}}{s}, \quad s = \frac{a+b+c}{2}

Cases where AAS is not applicable

For AAS two valid angles are required and there must be room for a third angle. So both entered angles must be greater than 0 degrees and the sum of the two angles must be less than 180 degrees. If the sum of the two angles is 180 degrees or more then no triangle exists, in which case the tool will explicitly inform you instead of trying to guess something.

If no corresponding pairs of angles and sides are known, AAS is not the appropriate tool. If all three sides are known (SSS) or two sides and the included angle are known (SAS), then the Law of Cosines must be used. If two sides and an opposite side of one of those sides are known (SSA), this is an ambiguous case where there may be two possible triangles, so the tool to use for calculation in this case would be the Law of Sines.

Practical examples of AAS

With the conditions of angles and sides, measurable angles can be converted into distances that are not directly accessible. Surveyors and sailors measure two angles on a distant target then determine the distance by measuring a baseline and solving for the triangle.

The same method is used in engineering for trusses, in astronomy for parallax measurements and all trigonometric problems where only two sight lines and a measured length are known.

Symbols explained:

The table below shows the individual symbols used in the above formulas.

Symbol	Meaning	Example
a, b, c	The three side lengths	16, 10.52, 22.56
A, B, C	The angles opposite a, b, c	40, 25, 115 deg
h_a, h_b, h_c	Heights (altitudes) onto a, b, c	9.53, 14.50, 6.76
R	Circumradius (circle through the vertices)	12.45
r	Inradius (largest inscribed circle)	3.11
s	Semi-perimeter, (a + b + c) / 2	24.54

This tool is for general learning and problem solving purposes. If the triangle has three sides given (SSS) or two sides and an included angle given (SAS), please use the Law of Cosines calculator. If two sides and a non-included angle are given (SSA, ambiguous case), please use the Law of Sines calculator.

Frequently asked questions

What is an AAS triangle?: AAS (Angle-Angle-Side) refers to a triangle in which two angles and one side are known, with the known side being opposite to one of the known angles. Since the third angle is determined by the two known angles (C = 180 - A - B), and the overall size of the triangle is fixed by the known side, an AAS triangle is always uniquely defined. This means that there is exactly one triangle that satisfies these conditions.
How to solve an AAS triangle?: First calculate the third angle by subtracting the sum of the two known angles from 180 degrees. Then use the law of sines to determine a common factor based on the known sides and their opposite angles. This common factor is then multiplied by the sine values of the other angles to find the lengths of the remaining sides. Finally, calculate the area, perimeter, and altitudes.
What is the difference between AAS and ASA?: Both have the condition of two angles and one side. In AAS, the known side is opposite to one of the known angles, so it's a non-included side. In ASA, the known side is between the two angles, so it's an included side. The solution methods are the same, and since both give exactly one triangle, they can be worked with this tool. You just have to choose which side is known.
What is the area of an AAS triangle with a 40-degree angle, a 25-degree angle and a side length of 16 centimeters?: For a triangle with angle A = 40 degrees, angle B = 25 degrees and side a = 16 cm, the third angle is C = 115 degrees. The formula for area is Area = 1/2*a squared*Sinus(B) * Sinus(A+B) / Sinus(A)" results in an area of approximately 76.27 square centimeters. The remaining two sides are each approximately b = 10.52 cm and c = 22.56 cm long.
How to calculate the height of an AAS triangle?: First calculate the missing angle C = 180 - A - B. The height to a side is twice the area divided by that side. With an equivalent formula this can be calculated in one step: the height to side b is a * Sin(C). Before plugging values in, make sure all angles are in the same units, either degrees or radians.
Can any AAS triangle be solved?: An AAS triangle can only be solved if both input angles are greater than 0 degrees and their sum is less than 180 degrees. This is because the third angle must be positive. If the sum of the two angles is greater than or equal to 180 degrees, then no triangle exists, and the tool will output a corresponding message instead of an invalid result. For any valid combination of angles with positive lengths, there is exactly one triangle that can be formed.
Can this tool be used to determine what type of triangle it is?: Yes it is possible. Based on the angles a tool can classify a triangle as acute triangle(all angles are less than 90 degrees), right triangle(one angle is 90 degrees) or obtuse triangle(one angle is greater than 90 degrees). Also based on the length of sides a tool can determine if a triangle is scalene triangle, isosceles triangle (two angles are equal) or equilateral triangle(all angles are 60 degrees).

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Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

Math is Fun: Solving AAS Triangles
Step-by-step method for solving angle-angle-side triangles with the law of sines.
NIST Special Publication 811: Guide for the Use of the International System of Units
The US national standard for SI units, unit names and symbols, and conversion factors.