How this instrument works
AAS names the three measurements you start from: two angles and a side, where the side sits opposite one of the given angles rather than trapped between them. Once two angles are known the third is not really new information — every triangle's angles sum to 180°, so C falls out at once as 180° − A − B. What actually does the work is the Law of Sines, a⁄sinA = b⁄sinB = c⁄sinC, which turns the one measured side into the other two once every angle is in hand.
The derivation is short: drop a perpendicular from one vertex onto the opposite side and it can be written two ways, once from each adjacent angle, as b·sinA and a·sinB. Set those equal and divide, and a⁄sinA = b⁄sinB appears with almost no machinery. The relation held for medieval astronomers long before it reached a Western trigonometry textbook — Nasir al-Din al-Tusi set out the general triangle case in the 13th century, computing star positions rather than classroom triangles.
The genuinely surprising part is that AAS is not a distinct piece of mathematics from ASA, the case where the known side sits between the two angles. Both reduce to the identical calculation the instant the third angle is found, because knowing two angles already fixes the whole shape before any side enters the picture — the label only records which side a surveyor happened to be able to reach with a tape. Push either given angle toward 0° and the triangle degenerates to a sliver; push their sum toward 180° and it flattens to a line, the one input this sheet refuses.
- Enter the two known angles into Angle A and Angle B, in degrees, radians, or turns — pick the unit from each field's menu.
- Enter the length of the side directly across from Angle A into the Side a (opposite A) field; this is the side that is not sandwiched between A and B.
- Read Angle C (computed): the sheet fills it in automatically as 180° minus the two angles you gave.
- Read Side b and Side c for the remaining lengths, both derived from Side a through the Law of Sines.
- Keep Angle A and Angle B summing to less than 180° — the sheet flags a pair that leaves no room for a third angle.
Worked example — angles of 30° and 60°, side a = 5
A drone survey fixes two bearing angles at a site: Angle A = 30° and Angle B = 60°, and a laser rangefinder returns the length of the side opposite A as 5 — a measurement the crew could take a clear line-of-sight on, unlike the side actually pinched between the two bearings. The third angle follows first: C = 180° − 30° − 60° = 90°, so the site is a right triangle without anyone having measured a right angle directly.
The remaining sides come from the Law of Sines. Side b = a·sinB ⁄ sinA = 5 × sin(60°) ⁄ sin(30°) = 5 × 0.8660254037844386 ⁄ 0.5 = 8.660254037844387. Side c = a·sinC ⁄ sinA = 5 × sin(90°) ⁄ sin(30°) = 5 × 1 ⁄ 0.5 = 10.000000000000002 — that trailing 2 is ordinary floating-point noise eighteen digits down, since a computer's sin(30°) sits a hair's breadth below the true value 0.5 rather than landing on it exactly. The clean answer underneath is the familiar 1 : √3 : 2 ratio — 5, 8.660254…, 10 — arrived at here from two bearings and a rangefinder rather than from a known side ratio.
Questions
What does AAS mean when solving a triangle?
Angle-Angle-Side: you know two of the triangle's angles and the length of a side, and that side is opposite one of the given angles rather than sitting between them. Once both angles are known the third follows by subtracting from 180°, and the Law of Sines then converts the one measured side into the other two.
How is AAS actually different from ASA, since both start from two angles?
Only in which side got measured. ASA's known side sits between the two given angles; AAS's sits opposite one of them. Mathematically the two cases collapse into the same calculation the moment the third angle is found by subtraction, because two known angles already fix the triangle's shape regardless of which remaining side you happened to measure.
Why doesn't AAS produce the two-answer ambiguous case that SSA does?
Because both angles are given outright, the triangle's shape is fixed before any side enters the calculation — there is nothing left to be ambiguous about. The SSA ambiguous case arises specifically when only one angle is known and the Law of Sines must solve for a second one, which can sometimes have two valid values between 0° and 180°.
What happens if the two given angles add up to 180° or more?
There is no third angle left to assign, so no triangle exists with those measurements; this sheet flags that input rather than returning a nonsense negative angle for C. Every planar triangle's three angles sum to exactly 180°, which is why any two of them must leave a positive remainder for the third.
I measured the side between the two angles instead — can I still use this?
That is the ASA case rather than AAS, and it uses a different pairing in the Law of Sines because the known side and the two given angles sit in a different arrangement around the triangle. The separate ASA Triangle Calculator on this site takes that included side directly.
How accurate are the sides this calculator returns?
As accurate as floating-point trigonometry allows — typically correct to about fifteen significant digits, with any remaining discrepancy sitting far below the precision of a tape measure or laser rangefinder. The tiny trailing digits sometimes visible in a result, such as a 2 appearing eighteen places after a value that is mathematically exactly 10, are that floating-point limit, not a flaw in the Law of Sines itself.