Area of a triangle

Where the half comes from

Every triangle is exactly half of a rectangle with the same base and height.

Take any triangle, make a copy, rotate it half a turn and slot it against the original. The two fit together into a rectangle of width $b$ and height $h$.

$$\text{rectangle} = bh \quad\Longrightarrow\quad \text{triangle} = \tfrac{1}{2}bh$$

The half is not an adjustment factor. It is there because you are looking at half a rectangle — and it holds for every triangle, not only right-angled ones.

What counts as the height

The height is the perpendicular distance from the base to the opposite corner. It is not a side of the triangle unless there is a right angle between them.

A sloping side is always longer than the perpendicular, so using one in its place overstates the area.

For a very slanted triangle the perpendicular lands outside the base, on its extension. The formula is unchanged — two copies still form the same rectangle.

Any side can be the base

A triangle has three base-and-height pairs, and all three give the same area. Use whichever pair the question hands you cleanly.

For a right triangle the two legs are already a pair, so no extra line needs drawing:

$$A = \tfrac{1}{2}(8)(6) = 24$$

Rearranged

$$b = \frac{2A}{h} \qquad h = \frac{2A}{b}$$

The $2$ appears because undoing the half means doubling first.

The signature error

Forgetting the half does not give a random wrong answer — it gives exactly the rectangle, twice the truth. An answer that is precisely double the expected one is this mistake every time. See area of a triangle.