Area of a Triangle
Geometry & MeasurementThe area of a triangle equals one-half times its base times its height, representing the region enclosed by its three sides.
Formula
A = \frac{1}{2} \times b \times h
Definition
The area of a triangle is half of the base times the height. You multiply the base length by the height and then divide by $2$.
Example
A triangle with a base of $10$ cm and a height of $6$ cm: $A = (1/2) \times 10 \times 6 = 30$ cm². The "half" comes from the fact that a triangle is exactly half of a rectangle with the same base and height.
Key Insight
Draw a rectangle around any triangle using its base and height. The triangle always fills exactly half the rectangle. That is why the formula has the $(1/2)$.
Definition
The area of a triangle with base $b$ and perpendicular height $h$ is $A = (1/2)bh$. This applies to all triangles, including obtuse triangles where the altitude foot falls outside the triangle. For a triangle with sides $a$, $b$, $c$ and semi-perimeter $s$, Heron's formula gives $A = \sqrt{s(s-a)(s-b)(s-c)}$.
Example
Triangle with sides $5$, $12$, $13$ (a right triangle): $A = (1/2)(5)(12) = 30$. Using Heron's: $s = 15$, $A = \sqrt{15 \cdot 10 \cdot 3 \cdot 2} = \sqrt{900} = 30$. Both methods agree.
Key Insight
Heron's formula is remarkable because it finds the area using only side lengths, without needing the height. It is especially useful in coordinate geometry and surveying.
Definition
For a triangle with vertices $(x_1,y_1)$, $(x_2,y_2)$, $(x_3,y_3)$, the signed area is $A = (1/2)\left|\det\begin{bmatrix}x_1&y_1&1\\x_2&y_2&1\\x_3&y_3&1\end{bmatrix}\right|$. The cross-product formula in 3-D: if two sides are vectors $u$ and $v$, the area is $(1/2)|u \times v|$.
Example
Vertices $(0,0)$, $(4,0)$, $(1,3)$: $A = (1/2)\left|\det\begin{bmatrix}0&0&1\\4&0&1\\1&3&1\end{bmatrix}\right| = (1/2)|0(0-3) - 0(4-1) + 1(12-0)| = (1/2)(12) = 6$.
Key Insight
The determinant formula for triangle area is the foundation of barycentric coordinates, computational geometry algorithms, and the shoelace formula for polygon areas, which extends the idea to any polygon.