Aaa Triangle Calculator

Background.

Three angles tell you everything about a triangle's shape and nothing about its size. Shrink a triangle on a photocopier or blow it up onto a billboard and the angles stay exactly as they were, while every side changes by the same factor. That is why angle-angle-angle is a test for similarity rather than congruence: triangles with matching angles are scaled copies of one another, but they need not be the same size.

What the angles do pin down is the proportion between the sides. By the law of sines each side is proportional to the sine of the angle opposite it, so a : b : c = sin A : sin B : sin C. Fix one side at 1 and the other two follow. Here the reference is side a, the side opposite angle A, and the results are b and c measured in units of a. The familiar 30-60-90 set square comes out as 1 : 1.732 : 2, which is 1 : √3 : 2.

Only two angles go in, because the third is whatever remains of 180°. Two angles that already use up 180° leave no room for a triangle, and the page refuses them.

To turn proportions into real lengths you need one measured side. Multiply every ratio by the true length of a, or, if the side you know is b or c, divide its length by its ratio to find the scale first. Perimeter grows in step with that scale factor and area with its square, which is why neither can be stated from the angles alone.

What is aaa triangle calculator?

An angle-angle-angle triangle is one described only by its three angles. Because the angles of a plane triangle always total 180°, two of them determine the third, and together they fix the triangle's shape but not its size. The side lengths can therefore be given only as proportions. Here they are normalized so that side a, opposite angle A, equals 1, and sides b and c are reported as multiples of it.

How to use this calculator.

1. 01Decide which side you will eventually measure and call the angle opposite it A, so that the side you know becomes the reference side a.
2. 02Enter angle A and angle B in degrees. Their sum must be less than 180°; angle C is worked out from what is left.
3. 03Read sides b and c as multiples of side a: a result of 1.732 means that side is 1.732 times as long as a.
4. 04To get real lengths, multiply both ratios by the measured length of a. If you measured side c instead, divide its length by the c ratio to get a, then scale b the same way.

The formula.

The third angle is C = 180° − A − B. The law of sines states that a ÷ sin A = b ÷ sin B = c ÷ sin C for every plane triangle. Setting a = 1 gives b = sin B ÷ sin A and c = sin C ÷ sin A. Degrees are converted to radians, multiplying by π ÷ 180, before the sines are taken. Angle A has the strongest influence on the answer because sin A sits in the denominator of both ratios: as A becomes small, sin A approaches zero and b and c grow without limit, reflecting a triangle whose reference side is tiny compared with the other two. With A = 1° and B = 60°, b is already about 49.6 times a.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, Rice University. Precalculus 2e, 2021. Trigonometry and analytic geometry chapters. Retrieved 2026-08-06.
2. [2]OpenStax, Rice University. College Algebra 2e, 2021. Algebra and function chapters. Retrieved 2026-08-06.
3. [3]NIST/SEMATECH. e-Handbook of Statistical Methods, 2012. Chapter 1.3 and Chapter 4. Retrieved 2026-08-06.