---
tag: 065JWMZW86
kind: concept
name: Area of a triangle
msc: [51M25]
uses: [0CTNFGYMM2]
sources:
  - url: https://en.wikipedia.org/wiki/Area_of_a_triangle
    locator: Basic formula
  - url: https://en.wikipedia.org/wiki/Area_of_a_triangle
    locator: Using coordinates
same_as:
  wikidata: Q20183185
  wikipedia: Area of a triangle
---

## Summary

The area of a [[triangle|0CTNFGYMM2]] is the measure of the region of the plane that the triangle bounds. It is half the product of a base and the height on that base: $T = \frac{1}{2} b h$.
Here $b$ is the length of a side, and $h$ is the distance from the opposite vertex to the line of that side.
If two sides have the lengths $a$ and $b$ and the angle $\gamma$ between them, then $T = \frac{1}{2} a b \sin \gamma$.
For the vertices $A = (x_A, y_A)$, $B = (x_B, y_B)$ and $C = (x_C, y_C)$ in $\mathbb{R}^2$:
$$T = \tfrac{1}{2} \left| (x_B - x_A)(y_C - y_A) - (x_C - x_A)(y_B - y_A) \right|.$$

## Remarks

[[Heron's formula|0FRBZ2X4HJ]] gives the area from the three side lengths alone.
In $\mathbb{R}^3$, the area of the triangle $ABC$ is $\frac{1}{2} \|(B - A) \times (C - A)\|$, with the [[cross product|0N5N15HWZ7]].
The coordinate formula without the absolute value is the signed area. It is positive when $A$, $B$, $C$ go round anticlockwise.

## Explanations

### L1 human

A triangle is half of a parallelogram: two copies of it fit together into one. The parallelogram has the area base times height. So the triangle has half of that, $\frac{1}{2} b h$.