CAT 2025 Lesson : Lines & Triangles - Area of a Triangle

3.4 Area of a Triangle

Following are the five ways to compute the area of a triangle.

Area	Figure
Area $= \dfrac{1}{2} b h$ where b is the base and h is the height or altitude of the triangle.
Area $= \sqrt{s(s - a)(s - b)(s - c)}$ where $a, b$ and $c$ are the lengths of the sides of the triangle and $s$ is the semi-perimeter, i.e.$s = \dfrac{ a + b + c}{2}$
Area $= \dfrac{1}{2} a b sin \theta$ where $\theta$ is the angle subtended by sides $a$ and $b$.
Area $= r \times s$ where $r$ is the inradius and $s$ is the semi-perimeter, i.e. $s = \dfrac{a + b + c}{2}$
Area $= \dfrac{abc}{4\mathrm{R}}$ where $a, b$ and $c$ are the lengths of the sides of the triangle and R is the circumradius.

3.5 Sine & Cosine Rules

Let us discuss some formulae linking triangles to trigonometry. In the below figure, A, B and C are the interior angles $\angle$CAB, $\angle$ABC and $\angle$BCA, and $a, b$ and $c$ are sides opposite these angles respectively.

Sine Rule: $\dfrac{a}{\mathrm {sin A}} = \dfrac{b}{\mathrm {sin B}} = \dfrac{c}{\mathrm {sin C}}$ Cosine Rule:  cos A $= \dfrac{ b^2 + c^2 - a^2}{2bc}$ cos B $= \dfrac{ c^2 + a^2 - b^2}{2ca}$ cos C $= \dfrac{ a^2 + b^2 - c^2}{2ab}$