Area and perimeter of a triangle

The area of a triangle is worked out almost exactly like the area of a rectangle — you just halve the result at the end. Where that half comes from is clearest once you draw the triangle's missing other half.

A rectangle with sides $a$ and $h$ has area $a \cdot h$. The diagonal cuts it into two congruent triangles — one is the other rotated — so their areas are equal. Hence:

The letter $a$ is the base, the side the triangle "stands on", and $h$ is the height dropped onto that base — the perpendicular segment from the opposite vertex.

Example: base 8 cm, height 5 cm

The height is not "the other side"

This is the most common mistake in the topic. The height has to be perpendicular to the base — in an obtuse triangle it can even fall outside the figure, and the formula still takes its length, not the length of a side.

A triangle has three bases and three matching heights. All three pairs give the same answer, so use the pair you actually know:

The right triangle: the easiest case

In a right triangle the two legs are already perpendicular to each other. One of them is the base and the other is a ready-made height, with nothing to draw in.

For legs of $3\,\text{cm}$ and $4\,\text{cm}$ the area is $\tfrac{1}{2} \cdot 3 \cdot 4 = 6\,\text{cm}^2$. The hypotenuse takes no part in it — its moment comes with the Pythagorean theorem.

Perimeter: simply the sum of the sides

The perimeter is the length of the line all the way round the triangle, so we add the three sides:

Not every three segments make a triangle. The triangle inequality has to hold: each side is shorter than the sum of the other two.

Sticks of length $2$, $3$ and $9$ will never form one — the two shorter ones simply do not meet.

Units: square for area, linear for perimeter

An area comes from multiplying two lengths, so we measure it in $\text{cm}^2$ and $\text{m}^2$; a perimeter comes from adding lengths, so it stays in $\text{cm}$ and $\text{m}$. When converting units of area, the length factor is squared — $1\,\text{m}^2 = 10{,}000\,\text{cm}^2$ — and if you need actual values converted, our area converter will do it.

Before you compute anything, check one more thing: every measurement has to be in the same unit. A triangle with a base of $1\,\text{m}$ and a height of $40\,\text{cm}$ can only be worked out once they match — $100\,\text{cm} \times 40\,\text{cm}$ gives $A = 2000\,\text{cm}^2$.

Exercises

Type the answer together with its unit, e.g. 20 cm² or 18 cm.

Common mistakes

• Forgetting the half — $a \cdot h$ is the area of a rectangle; a triangle has half of that.
• Confusing the height with a side — the height is perpendicular to the base; a side usually is not.
• Pairing a height with the wrong base — $h_a$ goes with $a$, $h_b$ with $b$; the pair has to match.
• Area instead of perimeter (and back) — areas are multiplied and written in $\text{cm}^2$, perimeters are added and written in $\text{cm}$.
• Using the hypotenuse in the area of a right triangle — you multiply the two legs.
• A triangle that does not exist — check the triangle inequality before you start.