How this instrument works
Take any triangle, pick one side as the base b, and measure the perpendicular distance h from that side to the opposite vertex. The area is exactly half the area of the b-by-h rectangle: A = b·h ⁄ 2. Euclid proved the underlying fact around 300 BC — a triangle sitting on a base between two parallels is half the parallelogram on the same footing — and no better formula has been needed since.
The height is the part people get wrong. It is not a side of the shape unless the triangle happens to be right-angled at the base; it is the length of the perpendicular dropped from the apex onto the base line — extended, if necessary, past the corner for obtuse shapes. Any of the three sides can serve as the base, and each side-and-height pairing gives the same area, a handy cross-check when you can measure more than one.
Units matter only in that both lengths must agree before multiplying. This instrument handles that for you: enter the base in metres and the height in feet if that is what your tape measure produced, and the conversion happens before the arithmetic. The result reads out in cm², m², or ft² — a metre-by-metre right triangle, for instance, is exactly 0.5 m².
- Enter the base — any side of the triangle you choose — in the Base field, picking cm, m, or ft from its unit menu.
- Measure the perpendicular height from that side to the opposite vertex and enter it in the Height field; its unit menu is independent of the Base field's.
- Read the result in the Area field, switching between cm², m², and ft² without re-entering anything.
- If a warning appears, check for a zero or negative entry — real triangles need both dimensions above zero.
Worked example — a 10 m by 6 m gable end
A shed's triangular gable end spans a 10-metre base, and the ridge rises 6 metres above it. Area: A = 10 × 6 ⁄ 2 = 60 ⁄ 2 = 30 m². That is the figure to hand the painter — two coats over 30 m² of timber, not the 60 m² a rectangle of the same outline would need.
The half is worth seeing once: copy the gable, rotate the copy half a turn, and the two pieces fit together into a 10 × 6 rectangle of exactly 60 m². One triangle is therefore 30 m² — the division by two is geometry, not convention.
Questions
Which side of the triangle is the base?
Any of them — the base is a choice, not a property of the shape. Pick whichever side you can measure cleanly, then take the height perpendicular to it. All three pairings give the identical area, so if two of them disagree, one of your measurements is off.
Is the height one of the triangle's sides?
Only in a right triangle, where the two legs serve as each other's base and height. Everywhere else the height is an invisible line: the perpendicular dropped from the opposite vertex onto the chosen side's line. In obtuse triangles that foot can land outside the shape — extend the line and measure to it; the formula still holds.
What if I know the three sides but not the height?
Use Heron's formula: with s = (a+b+c) ⁄ 2, the area is √(s(s−a)(s−b)(s−c)). This sheet deliberately sticks to base times height over two, the pair of measurements you usually have on site. For a 3–4–5 right triangle both routes agree: Heron gives √(6·3·2·1) = 6, and 3 × 4 ⁄ 2 = 6.
Does A = b·h ⁄ 2 work for every triangle?
Yes — scalene, isosceles, equilateral, right, obtuse. The proof does not care about shape: every triangle is half of a parallelogram with the same base and height, and the parallelogram's area is b·h. What changes between shapes is only how easy the height is to measure directly.
Can I mix units, say a base in metres and a height in feet?
Yes. Each input carries its own unit menu, and both entries are converted to a common unit before the multiplication, so a 10 m base with a 6 ft height computes correctly. Note that switching the Area field to ft² rescales by the square of the length factor — one square metre is about 10.764 ft², not 3.281.