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.

AAA Triangle Calculator

Enter three angles that sum to 180 degrees, plus one known side length, to calculate all sides, perimeter, area, height, inradius and circumradius using the Law of Sines, or classify the triangle's shape from angles alone.

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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

Angle A (degrees)

Angle B (degrees)

Angle C (degrees)

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)

Which side is known?

Side length (any unit)

units

Angle sum check

A + B + C = 180°

Triangle Results

Triangle Type

Equilateral

All angles equal

Area
43.30
sq units

Perimeter

30.00

units

Circumradius (R)

5.77

Circumscribed circle

Side Lengths

Side a (opposite A)10.00 units

Side b (opposite B)10.00 units

Side c (opposite C)10.00 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 sum180.00°

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:

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

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

Classification	Condition	Example
Acute	All angles less than 90°	60°, 70°, 50°
Right	One angle equals exactly 90°	90°, 45°, 45°
Obtuse	One angle greater than 90°	120°, 40°, 20°
Equilateral	All three angles equal 60°	60°, 60°, 60°
Isosceles	Exactly two angles are equal	70°, 70°, 40°
Scalene	All three angles are different	50°, 60°, 70°

Worked Example

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

Check: 60 + 60 + 60 = 180°. Valid.
Law of Sines: 2R = a / sin(A) = 10 / sin(60°) = 10 / 0.8660 = 11.547 units.
Side b = 2R × sin(B) = 11.547 × sin(60°) = 11.547 × 0.8660 = 10.00 units.
Side c = 2R × sin(C) = 11.547 × sin(60°) = 10.00 units.
Perimeter = 10 + 10 + 10 = 30.00 units.
Area = (1/2) × a × b × sin(C) = 0.5 × 10 × 10 × sin(60°) = 0.5 × 10 × 10 × 0.8660 = 43.30 sq units.
Height h_a = 2 × Area / a = 2 × 43.30 / 10 = 8.66 units.
Circumradius R = a / (2 × sin(A)) = 10 / (2 × 0.8660) = 5.77 units.
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

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Triangle Calculator (general): solve any triangle with any mix of sides and angles.
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SOHCAHTOA Solver: find missing sides and angles in right-angled triangles.
Right Angle Triangle Calculator: Pythagoras and trigonometry for right triangles.

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.