AAA Triangle Calculator

AAA" is a notation used when all three angles of a triangle are known. The most common case is where two angles are given and the third angle has to be calculated. This is easy to calculate since the sum of the interior angles in any triangle is always 180 degrees, so only one subtraction needs to be done.

However, AAA also reveals deeper geometric principles. While three angles completely determine the shape of a triangle, they do not fix its size. Two triangles are similar if their corresponding three angles are all equal respectively, and one is just an enlarged or reduced version of the other. This tool allows you to calculate unknown angles, identify the type of triangle, and display exact side ratios. If you enter the length of one side, the entire triangle can be determined with actual dimensions.

Formula for AAA to find third angle:

In a triangle with interior angles A, B and C, the sum of the three angles is always equal to 180 degrees.

If the angles are in radians, then the same relationship holds true: C = π - A - B. Calculating the third angle is simple; subtracting the two known angles from 180 degrees gives you the value of the third angle. If the sum of the two entered angles already exceeds 180 degrees, there is no room for a third angle and thus no triangle can be formed.

Example calculation

If the first two angles are 50 degrees and 60 degrees respectively, then the third angle is 180 - 50 - 60 = 70 degrees. Since all three angles are less than 90 degrees, it is an acute triangle. Also, since all three angles are different, it is a scalene triangle.

Why only shape is determined by the AAA but not size:

Even if all three angles are known, only the shape of the triangle is determined, not its size. A triangle with angles 50 degrees, 60 degrees and 70 degrees can be infinitely large or small, from a tiny sketch to an object as big as a plot of land, but always having the same shape. This is why we have the Angle-Angle (AA) similarity theorem: if two corresponding angles of two triangles are equal, then the third corresponding angles are also equal and the triangles are similar.

Since the shape is fixed, so are the ratios of each side. The sine rule can be used to directly calculate this ratio. The length of any side is proportional to the sine value of its opposite angle.

The calculator normalizes the shortest side to 1 and displays the ratio of each side so you can see by what factor the other two sides are larger than the smallest one. Also note that the side opposite the largest angle is always the longest, and that sides opposite equal angles are also equal in length.

Classifying triangles by their angles alone.

A triangle can be classified in two ways, by its angles or by its sides.

The classification by sides may seem surprising. Even if no side is measured, the angles determine whether a triangle is equilateral, isosceles or scalene. This shows how shape alone can be defined by the angles (AAA).

Determine actual size of a triangle given one side.

Because the size of an AAA triangle cannot be uniquely determined, a calculation tool requires the length of one side to define the dimensions. Enter any known side (a, b or c) and leave the other two sides blank. When one side and three angles are known, the remaining two sides can be calculated using the law of sines.

A common ratio of proportionality is determined from the known side and its opposite angle, and this ratio is used to calculate the length of either of the other sides.

The constant k is the same for all three sides and equals 2R, where R is the radius of the triangle's circumcircle. Once the lengths of the sides are known, other commonly used geometric quantities can be calculated.

Example calculation (if one side is given):

Since angle A = 40 degrees and angle B = 60 degrees, then angle C = 80 degrees. Enter side c = 10. The proportionality factor is k = 10 / sin 80 = 10.154, which gives a = 6.527 and b = 8.794. The calculated area is approximately 28.26, the perimeter is approximately 25.32, the circumradius is approximately 5.08, and the inradius is approximately 2.23.

Difference between AAA and conditions that uniquely determine a triangle:

Only the condition of AAA allows for an infinite number of similar triangles to be drawn, rather than a unique triangle; at least one side length must be known to define a specific triangle. The following table shows the difference between AAA and other common triangle congruence conditions.

For that reason, AAA is a similarity condition but not a congruence condition. With AAA it can be proved that two triangles have the same shape, but not that they are of the same size, or scale.

Examples of applications for the concept of AAA.

Similar triangles are a fundamental principle that underlies many everyday surveying methods. Surveyors and navigators measure the angles between distant landmarks and use fixed shapes and proportions to estimate distances that cannot be measured directly. Architects and designers can enlarge or reduce drawings by a given scale factor because the angles and ratio of sides remain unchanged regardless of scaling.

Photographers and optometrists use similar triangles to determine the path light takes through a lens. In fact, all scale models, from floor plans to maps, contain the principle of AAA. The angles remain the same, measurements may change but the shape remains intact.

Used symbols:

This calculator is for learning and solving problems only. As three angles alone determine the shape of a triangle, length of sides, area and perimeter will be shown only if one side length is entered.