Area of Triangles Using Sine and Heron's Formula Cheat Sheet

Key Facts

• The standard triangle area formula is $K = \frac{1}{2}bh$, where $b$ is the base and $h$ is the perpendicular height.
• The sine area formula is $K = \frac{1}{2}ab\sin C$, where $C$ is the included angle between sides $a$ and $b$.
• Equivalent sine area forms are $K = \frac{1}{2}bc\sin A$, $K = \frac{1}{2}ac\sin B$, and $K = \frac{1}{2}ab\sin C$.
• Heron's formula is $K = \sqrt{s(s-a)(s-b)(s-c)}$, where $a$, $b$, and $c$ are the side lengths.
• The semiperimeter is $s = \frac{a+b+c}{2}$, which is half of the triangle's perimeter.
• Before using Heron's formula, the side lengths must satisfy the triangle inequality: $a+b>c$, $a+c>b$, and $b+c>a$.
• If the included angle is $90^\circ$, then $K = \frac{1}{2}ab\sin 90^\circ = \frac{1}{2}ab$ because $\sin 90^\circ = 1$.
• Area units are always square units, such as $\text{cm}^2$, $\text{m}^2$, or $\text{in}^2$.

Vocabulary

Area

The area of a triangle is the amount of flat space inside it, measured in square units.

Included Angle

The included angle is the angle formed by two given sides of a triangle.

Sine

In triangle area problems, $\sin \theta$ connects an angle to the perpendicular height created from a side.

Semiperimeter

The semiperimeter is half the perimeter of a triangle, given by $s = \frac{a+b+c}{2}$.

Heron's Formula

Heron's formula finds triangle area from three side lengths using $K = \sqrt{s(s-a)(s-b)(s-c)}$.

Triangle Inequality

The triangle inequality says the sum of any two side lengths must be greater than the third side length.