How this instrument works
An oblique triangle simply means one with no right angle — but this sheet's real subject is a specific kind of data: one side, a=10 in the example below, flanked by two known angles, B and C. Because a triangle's angles always sum to 180°, the third angle falls out immediately as 180° minus the other two. The area formula then comes from the Law of Sines, a⁄sinA = b⁄sinB = c⁄sinC (the same ratio holds for every side, tied to the triangle's circumscribed radius): solving for side b as a·sinB⁄sinA and substituting into the ordinary Area = ½ab·sinC gives Area = a²·sinB·sinC⁄(2·sinA), where one measured side carries the whole answer once it is squared.
This is the ASA/AAS case of triangle-solving: two angles and a side, rather than three sides, which calls for Heron's formula, or two sides and the angle between them, which needs only a direct ½ab·sinC with no angle-sum step, or three vertex coordinates, which resolves through a shoelace determinant and no trigonometric ratio at all. Each combination of known measurements has one natural route to the area; forcing Heron's formula onto two angles and a side means inventing a missing side first, a detour this formula skips entirely.
The formula carries its own warning light. As B and C together approach 180°, the third angle shrinks toward zero, sinA shrinks with it, and the denominator heads toward nothing — the triangle is flattening into a straight line, and no finite area remains to report. Land surveyors meet this setup constantly, and in reverse from how it might seem: standing at one end of a measured baseline, sighting a distant point from both ends, and reading two angles off an instrument is far easier than pacing out a diagonal, which is exactly why ASA data, not three raw side lengths, is what triangulation naturally produces.
- Enter the one measured length into Side a — any unit works, as long as Area is read back in that same unit, squared.
- Enter the two angles that sit next to that side into Angle B and Angle C; the default unit is degrees, switchable to radians or turns.
- Angle A (computed) fills in on its own as 180° minus the two you entered — a quick way to sanity-check the pair before trusting the result.
- Read Area for the enclosed region, produced straight from the Law of Sines without ever constructing a missing third side.
Worked example — side 10 between angles 50° and 60°
Suppose a baseline runs a = 10 units between two sighted points, with the angle read at one end as B = 50° and at the other as C = 60°. The third angle follows immediately: 180° − 50° − 60° = 70°, which the sheet reports as Angle A (computed) = 70°, held internally as 1.2217304763960306 radians even though the display rounds the figure.
Feeding a = 10 with those same two angles into Area = a²·sinB·sinC ⁄ (2·sinA) gives 10² × sin(50°) × sin(60°) ⁄ (2 × sin(70°)), which works out to 35.299518879593556 square units — the exact figure this sheet returns, with no side ever measured directly and no rounding taken along the way.
Questions
Why does this formula need only one side instead of two or three?
Because the other two sides are already implied once two angles are fixed: shape is set by angles alone, and a single side fixes the scale. The Law of Sines turns that one measured side into a stand-in for both missing ones, so Area = a²·sinB·sinC⁄(2·sinA) needs nothing more than a, B, and C — feed it a=10, B=50°, C=60°, and it returns 35.3 without ever constructing side b or c explicitly.
How is this different from using Heron's formula?
Heron's formula wants three side lengths and no angles at all; this one wants a single side and two angles and no third side. When only ASA or AAS data exists — common in surveying and navigation, where angles come off an instrument rather than a tape measure — converting to three sides first would mean running a Law of Sines step anyway, so this route goes straight to the area instead.
What happens if the two given angles add up to 180° or more?
The triangle collapses: the third angle would be zero or negative, which describes no triangle at all, and the sheet's check flags this directly rather than returning a nonsense figure. Physically, B and C summing toward 180° means the two rays from the ends of side a are nearly parallel, stretching the shape flat before it disappears.
Does it matter which entered angle is called B and which is C?
No — sinB·sinC is symmetric, so swapping the two labels leaves the area unchanged; a=10 with B=50°, C=60° gives the same 35.299518879593556 as B=60°, C=50°. What does matter is that both angles are genuinely adjacent to side a rather than opposite it, since that adjacency is what the b = a·sinB⁄sinA step assumes.
Can the angles be entered in radians instead of degrees?
Yes — the Angle B and Angle C fields take degrees, radians, or turns; switch the unit selector and 50° becomes 0.8726646259971648 radians or about 0.1389 turn, all producing the identical 70° third angle and 35.3 area, since the trigonometry runs in radians internally regardless of what gets typed in.
What happens to the area if the one known side is doubled?
It quadruples rather than doubles, because area scales with a², the square of the one length supplied. Doubling a=10 to a=20 at the same B=50°, C=60° angles carries the area from 35.3 to 141.2 — four times as much, the same square-law every area formula follows when a single linear measurement is scaled up.