the area of a triangle
A triangle is exactly half of a parallelogram. Take any triangle, make a copy, flip the copy over, and fit the two together along a side: they snap into a parallelogram with the same base and height. Since the parallelogram has area b · h, one triangle has half of that.
The formula is A = (1/2) · b · h, where b is any side chosen as the base and h is the perpendicular height to that base — the distance from the opposite vertex straight down to the line of the base, measured at a right angle. You may use any of the three sides as the base, as long as you pair it with the matching height; all three choices give the same area. For an obtuse triangle the foot of the height can land outside the base, but the formula still holds.
If you do not know a height but you do know all three sides, Heron's formula gives the area directly. And if you know two sides and the angle between them, the trigonometric form A = (1/2) · a · b · sin C does the same job.
A triangle with base 8 cm and perpendicular height 5 cm has area (1/2) × 8 × 5 = 20 cm^2 — half of the 8-by-5 parallelogram you would get by doubling it.
Any side can be the base, paired with its own perpendicular height.
The height is the perpendicular distance to the base, not the length of a side; for an obtuse triangle this perpendicular may fall outside the triangle, yet the formula is unchanged.