AAA Triangle Calculator
Enter all three angles of a triangle to instantly get the relative side ratios and triangle classification. Add one known side length to unlock the actual side lengths, perimeter, area, inradius, and circumradius. The calculator validates that the angles sum to 180 degrees and shows a step-by-step breakdown of the Law of Sines solution.
What is an AAA triangle?
An AAA triangle is one where all three interior angles are known but no side length is given. Because the three angles of any triangle must sum to exactly 180 degrees, giving all three angles tells you the complete shape of the triangle - its proportions - but not its size. Think of two triangles with the same set of angles: one could fit on your thumbnail and another could span a field. They are similar (same shape, same proportions) but not congruent (different sizes). Mathematically, three angles define an equivalence class of infinitely many similar triangles. To pin down one specific triangle you need at least one side length alongside the angles.
What this calculator computes
Step one: the calculator checks that your three angles sum to 180 degrees (within a tiny rounding tolerance). If they do not, it flags the error and shows the actual sum. Step two: using the Law of Sines, it derives the relative side lengths. The Law of Sines states that a / sin(A) = b / sin(B) = c / sin(C), meaning the sides are proportional to the sines of the opposite angles. The calculator normalizes these ratios so the shortest side equals 1, making comparisons easy. It also classifies the triangle by both angle type (acute, right, or obtuse) and side type (equilateral, isosceles, or scalene). Step three: if you supply one known side length, the calculator scales the ratios to real dimensions and computes the other two sides, the perimeter, the area (via Heron's formula), the inradius (radius of the inscribed circle), and the circumradius (radius of the circumscribed circle).
The Law of Sines and why it works here
The Law of Sines is the key formula for AAA triangles: a / sin(A) = b / sin(B) = c / sin(C) = 2R, where R is the circumradius. When all three angles are known, the ratio of any two sides is fixed: b / a = sin(B) / sin(A). This means you can find every side relative to any one given side. For example, if angle A = 60 degrees, angle B = 80 degrees, and angle C = 40 degrees, then sin(60) = 0.8660, sin(80) = 0.9848, sin(40) = 0.6428. Normalizing by the smallest sine gives relative sides of a : b : c = 1.347 : 1.532 : 1.000. If you are then told side c = 10, all other sides and dimensions follow immediately from scaling.
Practical uses of AAA triangle analysis
Surveyors and navigators frequently know angles before they know distances: triangulation measures two angles from known baselines to locate a distant point. Architects and engineers use angle-defined triangular bracing where proportions matter first and sizing comes later. In photography and optics, angle of view calculations rely on triangular geometry where only angles are initially specified. In astronomy, parallax measurements begin with angles observed from different positions before converting to actual distances using a known baseline. Computer graphics and game engines use angle-specified similar triangles for level-of-detail scaling, fitting a pre-proportioned triangle into a scene by specifying just one dimension at render time.
Triangle classification quick reference
| Angle type | Largest angle | Side type | Side relationship |
|---|---|---|---|
| Acute | All angles < 90 | Equilateral | a = b = c (all angles = 60) |
| Acute | All angles < 90 | Isosceles | Two sides equal, two angles equal |
| Acute | All angles < 90 | Scalene | All sides and angles different |
| Right | Exactly one angle = 90 | Isosceles | Two legs equal (45-45-90) |
| Right | Exactly one angle = 90 | Scalene | Pythagorean triple or general |
| Obtuse | One angle > 90 | Isosceles | Base angles equal, apex > 90 |
| Obtuse | One angle > 90 | Scalene | All sides and angles different |
How triangles are classified by their angles and side relationships.
Frequently asked questions
Can you fully solve a triangle with only three angles?
Not completely. Three angles define the shape (relative proportions) of a triangle but not its scale. Infinitely many triangles share the same three angles - they are all similar to each other but differ in size. You can always find the relative side ratios using the Law of Sines (a : b : c = sin A : sin B : sin C), and classify the triangle type. To find actual side lengths, area, perimeter, inradius, or circumradius you also need at least one side length.
Why must the three angles sum to exactly 180 degrees?
This follows from the parallel postulate of Euclidean geometry. Draw a line through any vertex of a triangle parallel to the opposite side. The alternate interior angles show that the three interior angles fit together to form a straight angle (180 degrees). This holds for every flat (Euclidean) triangle without exception. On curved surfaces like a sphere, triangles can have angle sums greater than 180 degrees - a triangle on Earth drawn between the North Pole and two equatorial points can have angles summing to 270 degrees.
What is the Law of Sines and how does it apply here?
The Law of Sines states that each side of a triangle divided by the sine of the angle opposite to it gives the same constant: a / sin(A) = b / sin(B) = c / sin(C) = 2R, where R is the circumradius of the triangle. When all three angles are known, the ratios a : b : c = sin(A) : sin(B) : sin(C) are fixed. This means if you know any one side, you can find the others by proportional scaling.
What is the inradius and how is it calculated?
The inradius (r) is the radius of the largest circle that fits entirely inside the triangle, touching all three sides. It is calculated as r = Area / s, where s is the semi-perimeter (half the perimeter). The inscribed circle is always tangent to each side at exactly one point. A larger inradius relative to the triangle means a more "compact" or "fat" triangle.
What is the circumradius and when is it useful?
The circumradius (R) is the radius of the circle that passes through all three vertices of the triangle. It equals a / (2 sin A), which is the same as b / (2 sin B) or c / (2 sin C) by the Law of Sines. It appears frequently in trigonometry, navigation (circumscribed circles around waypoints), and engineering design where a triangular frame must fit within a given circle.
How do I use this to solve a triangle from a map bearing?
If you know two compass bearings from a fixed baseline (a common surveying setup), you have two angles. The third angle is 180 minus their sum. You then have an AAA triangle. If the baseline length is known - as it is in triangulation - enter it as the known side to get all actual dimensions. This is the classic triangulation method used in surveying, cartography, and GPS positioning.