🔺 Triangles

• Any triangle: $\text{Area}=\dfrac{1}{2}\times b\times h$, where $h$ is the height drawn to the base $b$.

• Equilateral (side $a$): area $\dfrac{\sqrt3}{4}a^2$, height $\dfrac{\sqrt3}{2}a$, perimeter $3a$.

• Right triangle: the two legs are the base and height, and Pythagoras gives $a^2+b^2=c^2$, so the hypotenuse is $c=\sqrt{a^2+b^2}$.

• Heron (three sides): with $s=\dfrac{a+b+c}{2}$, area $=\sqrt{s(s-a)(s-b)(s-c)}$.

• Ratio of areas: for similar triangles $\dfrac{A_1}{A_2}=\left(\dfrac{s_1}{s_2}\right)^2$ — the square of the side ratio. But if only the base (or only the height) changes, the area ratio equals that single ratio, not its square.

The exam habit to build: when the sides form a Pythagorean triple such as $5,12,13$ or $9,40,41$, do not reach for Heron — the triangle is right-angled, so the area is just half the product of the two legs.