Area of a triangle

Japanese school year: Math I

What you learn

Learn how to calculate triangle area using two sides and the sine of the included angle, as well as the inradius. This builds upon basic area concepts and trigonometric ratios to solve oblique triangles. It is widely used in land surveying, computer graphics, and decomposing polygons into measurable parts.

Go to practice

Key points

This formula calculates a triangle's area $S$ from two adjacent sides $a, b$ and their included angle $C$ . It provides the area using $\sin C$ without knowing the height.

S = \frac{1}{2}ab\sin C

This formula links triangle area $S$ , inradius $r$ , and side lengths $a, b, c$ . It is commonly used to solve for inradius $r$ when the area $S$ is already known.

S = \frac{1}{2}r(a + b + c)

Known as Heron's formula, this calculates area $S$ solely from three side lengths $a, b, c$ without needing angles. Here, $s$ is the semiperimeter $\frac{a+b+c}{2}$ .

S = \sqrt{s(s-a)(s-b)(s-c)} \quad \left(s = \frac{a+b+c}{2}\right)

Choose a set to practice.

Practice 1Coming soon