A triangle's area and perimeter from three sides, by Heron's formula.
With a base and a height, the area is S = ½ × base × height. The height must be perpendicular to the base you chose, not to just any side — that is the usual mistake. With a base of 8 cm and a height of 5 cm the area is 0.5 × 8 × 5 = 20 cm².
It gives the area from the three sides alone, with no height needed. First take the semi-perimeter p = (a + b + c) / 2, then S = √(p(p − a)(p − b)(p − c)). This is what you want when measuring land, where sides are easy to measure and a height is all but impossible.
In a right-angled triangle the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c². With legs of 3 and 4 the hypotenuse is √(9 + 16) = 5. It holds only for right angles, and it is also used in reverse — to check whether a corner really is square.
Only if the sum of any two sides exceeds the third. Lengths of 2, 3 and 10 form no triangle, because 2 + 3 is less than 10 — the two short sides simply cannot meet. This is the triangle inequality, and it is the first thing to check when a calculation returns something odd.