Start with opposite side-angle pairs
For triangle sides a,b,c opposite angles A,B,C, the law of sines is a/sin(A)=b/sin(B)=c/sin(C). A side and its opposite angle belong in the same ratio. A diagram's page orientation does not determine those labels. ASA supplies two angles and their included side; AAS supplies two angles and a non-included side. Both establish the third angle, after which a known side fixes the scale of the triangle rather than only its shape.
Find the third angle before scaling sides
Angles of a nondegenerate plane triangle sum to 180°. With A=30° and B=60°, C=90°. Require all angles to be positive and their known sum to be below 180°. A sum equal to 180° leaves a zero angle and does not describe a positive-area triangle. Two angles alone give similarity but no unique side lengths: triangles with the same angles can be enlarged without changing them. One positive side is therefore essential to these solving cases.
Work an ASA example with included side c
Take A=30°, B=60° and their included side c=10. Since C=90°, the common scale c/sin(C) is 10. Thus a=10sin(30°)=5 and b=10sin(60°)=5sqrt(3), approximately 8.660254038. The side between vertices A and B is opposite C; identifying that included side correctly is the main setup step. If c were mistakenly treated as a, the solution would have a different scale despite using the same angles.
Work the matching AAS example
Now supply A=30°, B=60° and opposite side a=5. The scale is a/sin(A)=5/0.5=10, so b=5sqrt(3) and c=10 again. This is AAS because the supplied side is not the side included between the two given angles. The two examples reproduce one triangle through different sufficient input cases. Their agreement is a useful cross-check, but it depends on matching the same labeled side; a length of 5 assigned to b would describe another size.
| Case | Supplied measurements | Recovered sides |
|---|---|---|
| ASA | A=30°, c=10, B=60° | a=5; b=5√3; c=10 |
| AAS | A=30°, B=60°, a=5 | a=5; b=5√3; c=10 |
Lowercase sides are opposite their matching uppercase angles.
Use degrees consistently
When evaluating spreadsheet or programming sine functions that expect radians, convert degrees with angle×π/180 before taking the sine. A dedicated degree-mode calculator can accept the original angle directly. Entering sin(30) into a radian-based function is not sin(30°). The Triangle Calculator's labeled angle inputs use degrees. Keep angle units with any copied formula, particularly when a sheet mixes trigonometric outputs with ordinary length arithmetic. A plausible side length is not enough to prove the angle convention was correct.
Distinguish ASA and AAS from SSA
With two known angles, the third is fixed and positive valid data determine one triangle up to congruence and reflection in the drawing. SSA instead supplies two sides and a non-included angle and can produce two triangles, one or none. Do not introduce the supplementary-angle branch into an ASA calculation whose angles are already specified. Conversely, do not apply a two-angle workflow to SSA by silently inventing a second angle. Choose the input case that reflects the supplied information.
Verify area and side ordering
In the example, the largest angle 90° is opposite the largest side 10, while 30° is opposite the smallest side 5. The area is half the product of the perpendicular legs: 5×5sqrt(3)/2=25sqrt(3)/2, approximately 21.65063509 square units. Perimeter is 15+5sqrt(3), approximately 23.66025404. These extra checks can reveal swapped labels. The angle sum alone cannot detect every error because incorrect side scales can coexist with perfectly valid angles.
Use the calculator's labeled modes
In ASA mode, the website asks for A, c and B. Enter 30,10,60 to recover the example. In AAS mode, it asks for A,B,a; enter 30,60,5. Confirm the labels after switching modes because default examples can change. The tool solves supplied geometry and does not infer a drawing's intended side labels. Lengths must use the same unit, and area will use its square. Very extreme angles or side ratios may fall outside supported numerical precision.
Report a solution without false exactness
Retain unrounded trigonometric values until final presentation and keep exact expressions such as 5sqrt(3) when appropriate. If angles were measured approximately, additional output decimals do not establish equally precise physical lengths. Check the result against the original side and angles before using it in another geometric task. For a real structure, measurement conventions, clearances and construction tolerances remain separate from the ideal triangle. A reviewable solution states the input case, labels, angle unit and all given measurements.