Area of Triangle

1. Introduction: Finding the Size of a Plot Without Stepping on It

Picture a triangular plot of farmland at the edge of a village. You know the three corners — you can mark them on a map with grid lines — but you cannot walk across the field with a measuring tape, because the crop is standing tall.

Here is the question this whole lesson answers: if I only know the three corners of a triangle as points on a grid, can I find its area without measuring any side or any height directly?

Earlier you learnt to find the distance between two points on a grid. You also learnt that a triangle's area is $\dfrac{1}{2} \times \text{base} \times \text{height}$. But finding the height of a slanted triangle on a grid is painful — you would have to drop a perpendicular and chase its length.

There is a cleaner way. It uses only the three corner points, plugged straight into one formula. By the end you will be able to take any three points, feed them in, and read off the area.

Stop scrolling. Think for a moment: if a triangle's three corners all sat on one straight line, what would its "area" be? Try to picture it before reading on.

A triangle whose corners lie on one straight line is flat — it has no inside at all. Its area is $0$. Hold on to that idea; it becomes a powerful test later in this lesson.

You can now say what this lesson is for: to find a triangle's area from its three corner points alone.