Triangle Area Calculator

The simplest formula is $A = \frac{1}{2} \times b \times h$, where $b$ is the base and $h$ is the perpendicular height. The height must form a 90-degree angle with the chosen base. This works for all triangles but requires knowing the perpendicular height.

Heron's formula calculates area from three side lengths: $A = \sqrt{s(s-a)(s-b)(s-c)}$, where $s = \frac{a+b+c}{2}$ is the semi-perimeter. For example, a triangle with sides 7, 8, 9 has $s = 12$ and area $\sqrt{12 \times 5 \times 4 \times 3} \approx 26.83$.

Use the SAS formula $A = \frac{1}{2} \times a \times b \times \sin(C)$ when you know two sides and the angle between them. It is the most direct method for this case, since the perpendicular height equals $b \sin(C)$.

The shoelace formula computes area from vertex coordinates: $A = \frac{1}{2}|x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$. The absolute value ensures a positive result regardless of vertex ordering. It is ideal when you have coordinate geometry data.

Three sides form a valid triangle if and only if the sum of any two sides exceeds the third (the triangle inequality). Equivalently, when computing Heron's formula, if any of $s-a$, $s-b$, or $s-c$ is negative or zero, the triangle is impossible.