Area of a triangle

What the symbols mean

$A$

the area

$b$

the base: any side you choose

$h$

the perpendicular height: the distance from the base (or its extension) to the opposite corner, at right angles to the base

When to use it

Use it when you know a side and the height measured at right angles to it. If you know three sides instead, use Heron's formula; if you know two sides and the angle between them, use $\frac{12}{a}b\sin C$.

Why it works

Any triangle is half of a parallelogram with the same base and height (put two copies together), and a parallelogram has area $bh$. In the graph, slide the top corner sideways: the base and height stay the same, so the area does not change.

Common mistakes

• Using a sloping side as the height. The height must be perpendicular to the base.
• Forgetting the ½: base × height is the area of the parallelogram, twice the triangle.
• Thinking an obtuse triangle has no height. It does; it falls outside the triangle, onto the extended base (slide the top corner past the end in the graph).