How this instrument works
An obtuse triangle has one interior angle strictly greater than 90 degrees, and the extra room that wide angle claims is exactly what pushes the altitude's foot outside the shape once you measure it from a vertex next to that angle. This calculator assumes the base and its perpendicular height are already known — not two sides and the angle between them, not three side lengths run through Heron's formula, not a set of coordinates — because base and height are the two figures a builder's square or a computed altitude hand you directly, and A = half of base times height turns them into an area with nothing else required.
Why the formula does not care where that foot lands: slide the opposite vertex sideways along a line parallel to the base, and neither the base's length nor the perpendicular distance to that parallel line changes, so the enclosed area cannot change either — the triangle is still trapped between the same base and the same parallel. Push the vertex far enough sideways and the foot of the perpendicular slips right past the end of the base onto that line's own extension; an obtuse triangle is simply one where the vertex has slid far enough for that to happen. Nothing in half-base-times-height ever required the foot to sit between the base's own two endpoints.
Watch what happens as that obtuse angle keeps opening, toward a straight 180 degrees: the base stays fixed, the perpendicular height shrinks toward zero, and the area collapses to nothing, exactly as a triangle flattening into a line should. On site, that displaced foot is the detail to plan for — a square held at the base's visible end checks a right angle that is not actually there, so the base line needs extending with a straightedge or a taut string before the perpendicular can be dropped and measured honestly.
- Enter the triangle's chosen side length into Base.
- Drop a true perpendicular from the opposite vertex to that base's line, extending it past its own end if needed, and enter the distance into Height (perpendicular, may fall outside the triangle).
- Read the result in Area: half of Base times Height.
- If Area looks too small, check whether Height was measured only to the base's visible edge rather than to the extended line.
Worked example — a 10-metre base with a 4-metre height
A surveyor stakes a triangular parcel using a 10-metre fence line as the base and finds a 4-metre perpendicular offset from the far corner to the line carrying that fence. Area: A = half of 10 times 4, which is 40 divided by 2, or 20 square metres — entered on the survey plat without a single interior angle or side length beyond the two figures actually measured.
That far corner sits well off to one side rather than squarely above the fence: the parcel's one obtuse angle falls at the western stake, so the true right-angle foot for the height lands two metres beyond that stake, past the fence's own end and onto a string extended along its line. Measuring only between the fence's two visible ends would never meet the foot at all; extending the line first, then dropping the perpendicular, is what makes both the 4-metre reading and the 20-square-metre answer correct.
Questions
Why does area equal half of base times height even when the foot lands outside the triangle?
Because area depends only on the base's length and the perpendicular distance between the base's line and a second line through the opposite vertex, parallel to the base — not on where along that second line the vertex actually sits. Slide the vertex sideways and both the base and the perpendicular distance stay fixed, so the area cannot change, even once the slide carries the foot past the base's own end.
Which vertex's angle decides which side the foot falls outside on?
Whichever base vertex holds the triangle's obtuse angle. Drop the altitude from the third vertex onto the base's line: if the left base vertex is the obtuse one, the foot lands beyond that same left vertex; an obtuse angle on the right pushes the foot beyond the right end instead. A triangle carries at most one obtuse angle, so only one end of the base can ever be affected.
How do I measure the height if extending the base line isn't practical?
Compute it rather than draw it. Given the base and the two other sides, Heron's formula gives the area directly, and solving A = half of base times height for height recovers the perpendicular distance without drawing it on the ground at all. A known angle at one base vertex, paired with the adjacent side, reaches the same height through simple trigonometry.
How does this differ from the SAS or three-sides area formulas?
Those two calculators start from measurements this one skips entirely: SAS needs two sides and the angle trapped between them, and the three-sides version runs Heron's formula on all three lengths with no height anywhere in sight. This calculator assumes the base and its perpendicular height are already known, the plainest pair of inputs a triangle offers, and needs no trigonometry or square roots to turn them into an area.
What happens to the area as the obtuse angle widens toward 180 degrees?
It shrinks toward zero. A wider obtuse angle flattens the triangle toward a straight line lying along the base, and the perpendicular height collapses toward zero even while the base itself stays fixed; half of base times height falls right along with it, exactly as a flattened triangle with no enclosed area should.
Does it matter which of the three sides I call the base?
No. Any side can serve as the base, provided the height entered is the perpendicular distance from the opposite vertex to that particular side's own line, extended past the shape when the adjoining angle is obtuse. All three base-and-height pairings return the identical area; if two of them disagree, one of the two measurements feeding this sheet is wrong.