Area of a Triangle — All Formulas with Examples

To find the area of a triangle, use the formula that matches the information you have. If the problem gives a base $b$ and the perpendicular height $h$, the main formula is

$$A = \frac{1}{2}bh$$

If the height is not given, you can still find the same area from two sides and an included angle, from all three side lengths, or from coordinates. The key is choosing the formula whose condition actually fits the triangle.

Why the triangle formula has $\frac{1}{2}$

A triangle with base $b$ and height $h$ has half the area of a rectangle or parallelogram built on the same base and height. That is why the factor $\frac{1}{2}$ appears.

The condition matters: $h$ must be perpendicular to the base you chose. A slanted side is not a height unless it meets the base at a right angle.

Area of a triangle formulas and when to use each one

Base and perpendicular height

Use this when a base and its corresponding height are known.

$$A = \frac{1}{2}bh$$

This is the most direct formula and usually the fastest one.

Two sides and the included angle

Use this when you know sides $a$ and $b$ and the angle $C$ between them.

$$A = \frac{1}{2}ab\sin C$$

This works because the height relative to side $b$ is $a\sin C$.

Heron's formula

Use this when you know all three sides $a$, $b$, and $c$, but not the height.

$$s = \frac{a+b+c}{2}$$ $$A = \sqrt{s(s-a)(s-b)(s-c)}$$

Here, $s$ is the semiperimeter. This formula is useful when the side lengths are known but no angle or altitude is given.

Coordinate formula

Use this when the triangle is given by points $(x_1,y_1)$, $(x_2,y_2)$, and $(x_3,y_3)$ in the coordinate plane.

$$A = \frac{1}{2}\left|x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)\right|$$

The absolute value is important because area should not be negative.

Equilateral triangle formula

Use this only when all three sides are equal and each side has length $a$.

$$A = \frac{\sqrt{3}}{4}a^2$$

This is a special case, not a general triangle formula.

Worked example: area of a $3$-$4$-$5$ triangle

Suppose a triangle has side lengths $3$, $4$, and $5$. Since $3^2 + 4^2 = 5^2$, it is a right triangle, so the sides of lengths $3$ and $4$ are perpendicular. That makes them the easiest base and height to use.

Let $b = 4$ and $h = 3$.

$$A = \frac{1}{2}bh = \frac{1}{2}(4)(3) = 6$$

So the area is $6$ square units.

If you want a check, Heron's formula gives the same result:

$$s = \frac{3+4+5}{2} = 6$$ $$A = \sqrt{6(6-3)(6-4)(6-5)} = \sqrt{36} = 6$$

The lesson is not that you should use every formula each time. The lesson is that different formulas give the same area when their conditions are satisfied.

Common mistakes with triangle area

The most common mistake is using a side length as the height without checking that it is perpendicular to the chosen base.

Another mistake is using $A = \frac{1}{2}ab\sin C$ with an angle that is not between sides $a$ and $b$. In that formula, the angle must be the included angle.

In Heron's formula, students often forget to compute the semiperimeter first or mix up $s$ with the full perimeter. Small arithmetic errors also matter because everything is inside a square root.

For coordinate problems, forgetting the absolute value can produce a negative number, which cannot be an area.

When each triangle area formula is useful

Use $A = \frac{1}{2}bh$ in basic geometry, construction sketches, and any problem where the altitude is easy to see or compute.

Use $A = \frac{1}{2}ab\sin C$ in trigonometry and surveying-style problems where two sides and an angle are known.

Use Heron's formula when all three side lengths are known and introducing the height would be awkward.

Use the coordinate formula in analytic geometry, graph problems, and cases where the triangle is defined by vertices instead of side-height data.

Use the equilateral formula only when the triangle is equilateral. If the triangle is merely isosceles, that shortcut does not automatically apply.

How to choose the right formula fast

If you know base and perpendicular height, use $A = \frac{1}{2}bh$.

If you know two sides and the angle between them, use $A = \frac{1}{2}ab\sin C$.

If you know all three sides, use Heron's formula.

If you know the coordinates, use the coordinate formula.

If the triangle is equilateral, the special shortcut is available.

Try a similar problem

Try your own version with a triangle whose sides are $5$, $12$, and $13$. First notice what kind of triangle it is, then find the area in the fastest way. After that, solve it again with Heron's formula and check that both answers agree.