AAA Triangle Calculator

This calculator solves a triangle when you know all three interior angles (Angle-Angle-Angle, or AAA) and, optionally, the length of one side. Three angles alone fix a triangle's shape but not its size, so you enter Angle A, Angle B and Angle C in degrees (they must sum to exactly 180 degrees), then choose which side is known, side a, b or c, and its length, or leave it as ratios only if no side measurement is available. Using the Law of Sines (a/sin A = b/sin B = c/sin C), the calculator works out the two remaining side lengths, then returns the triangle type (acute, right or obtuse, and equilateral, isosceles or scalene), the area, perimeter and circumradius as headline results, plus a detailed breakdown of all three side lengths, the side ratio, the height to each side, the inradius and a check on the angle sum. A plain-language summary underneath describes the shape and dimensions. Use it to check homework, verify a set of surveying or triangulation angles, or see how a triangle's dimensions change as you adjust its angles or known side. Without a side length you still get the triangle's classification and the ratio between its sides, since AAA fixes the shape but never the size. Results are exact for flat Euclidean triangles and are rounded to two decimal places for display.

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Standard Formula  Law of Sines: a/sin(A) = b/sin(B) = c/sin(C). Area: (1/2) × a × b × sin(C).

1. Three Angles

°
°
°
Angles must each be between 0° and 180° (exclusive) and sum to exactly 180°.
AAA defines the triangle shape only. Enter a side below to find actual dimensions.

2. Known Side (Optional)

units
A + B + C = 180°

Triangle Results

Triangle Type
Equilateral
All angles equal 60° / Equilateral
Area
43.3
sq units
Perimeter
30
units
Circumradius (R)
5.77
Circumscribed circle

Side Lengths

Side a (opposite A)10 units
Side b (opposite B)10 units
Side c (opposite C)10 units
Ratio a : b : c1 : 1 : 1

Other Properties

Height to side a (h_a)8.66 units
Height to side b (h_b)8.66 units
Height to side c (h_c)8.66 units
Inradius (r)2.89 units
Angle sum180°
Summary: An equilateral triangle has all three angles equal to 60°. All sides are equal. With side a = 10 units, each side is 10.00 units, the perimeter is 30.00 units, and the area is 43.30 sq units.

What is AAA in Triangle Geometry?

AAA stands for Angle-Angle-Angle, the case where all three interior angles of a triangle are known. The angles of any triangle always sum to exactly 180°, so knowing two angles is enough to determine the third: C = 180° - A - B.

The key limitation of AAA is that it does not uniquely determine the size of the triangle. It only determines the shape. All triangles with the same three angle measures are similar (proportional) to each other. To find actual side lengths, area, and perimeter, you need at least one side length in addition to the three angles.

Law of Sines

Once one side is known, the Law of Sines relates all three sides to their opposite angles:

a / sin(A) = b / sin(B) = c / sin(C) = 2R

where R is the radius of the circumscribed circle (the circle that passes through all three vertices). From the known side, you calculate 2R, then find the other sides by multiplying 2R by the sine of the opposite angle. For example, if side a is known:

Area from Two Sides and Included Angle

Once all three sides are known, the area can be calculated using any two sides and their included angle:

Area = (1/2) × a × b × sin(C)

Equivalently, Area = (1/2) × b × c × sin(A) = (1/2) × a × c × sin(B). All three give the same result.

Triangle Classification

ClassificationConditionExample
AcuteAll angles less than 90°60°, 70°, 50°
RightOne angle equals exactly 90°90°, 45°, 45°
ObtuseOne angle greater than 90°120°, 40°, 20°
EquilateralAll three angles equal 60°60°, 60°, 60°
IsoscelesExactly two angles are equal70°, 70°, 40°
ScaleneAll three angles are different50°, 60°, 70°

Worked Example

Default inputs: A = 60°, B = 60°, C = 60°, side a = 10 units.

  1. Check: 60 + 60 + 60 = 180°. Valid.
  2. Law of Sines: 2R = a / sin(A) = 10 / sin(60°) = 10 / 0.8660 = 11.547 units.
  3. Side b = 2R × sin(B) = 11.547 × sin(60°) = 11.547 × 0.8660 = 10.00 units.
  4. Side c = 2R × sin(C) = 11.547 × sin(60°) = 10.00 units.
  5. Perimeter = 10 + 10 + 10 = 30.00 units.
  6. Area = (1/2) × a × b × sin(C) = 0.5 × 10 × 10 × sin(60°) = 0.5 × 10 × 10 × 0.8660 = 43.30 sq units.
  7. Height h_a = 2 × Area / a = 2 × 43.30 / 10 = 8.66 units.
  8. Circumradius R = a / (2 × sin(A)) = 10 / (2 × 0.8660) = 5.77 units.
  9. Inradius r = Area / s, where s = perimeter / 2 = 15. r = 43.30 / 15 = 2.89 units.

This matches the default output shown in the calculator above.

Related Calculators

Sources and method: Law of Sines (Trigonometry, standard curriculum). Area formula: (1/2)ab sin C. Inradius: r = Area / s where s is the semi-perimeter. Circumradius: R = a / (2 sin A). Triangle classification follows standard Euclidean geometry definitions.

This calculator provides exact results for Euclidean (flat) triangles. All angles must be strictly between 0° and 180° and must sum to exactly 180°. Results are rounded to two decimal places for display. The area and side results require at least one known side length; without a side, only shape classification and side ratios are available.

Frequently asked questions

Can you find the sides of a triangle from angles alone (AAA)?

Not without at least one side length. Three angles (AAA) define the shape and proportions of a triangle but not its size. Any triangle with the same three angles is similar to yours but could be any scale. Once you know one side, the Law of Sines determines all remaining sides: a/sin(A) = b/sin(B) = c/sin(C). This ratio is called the circumscribed circle diameter (2R). Without a side, you can still classify the triangle (acute, right, or obtuse; scalene, isosceles, or equilateral) and find the ratios between the sides.

What does AAA mean in triangle geometry?

AAA stands for Angle-Angle-Angle, meaning all three interior angles of a triangle are known. In geometry, AAA is a triangle similarity criterion: if two triangles have the same three angle measures, they are similar (same shape, proportional sides). AAA is not a congruence criterion because it does not fix the size of the triangle. By contrast, SSS (three sides), SAS (two sides and included angle), and ASA or AAS (two angles and a side) are all congruence criteria that fully determine a unique triangle.

How do I classify a triangle from its angles?

A triangle is classified by its largest angle. If all angles are less than 90°, it is acute. If one angle equals exactly 90°, it is a right triangle. If one angle is greater than 90°, it is obtuse. By side lengths (or equivalently by angle equality): if all three angles are equal (each 60°) the triangle is equilateral; if exactly two angles are equal it is isosceles; if all three differ it is scalene. Note that an equilateral triangle is always also equiangular, and a right triangle can be isosceles (the 45-45-90 triangle) but never equilateral.