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Area of a Triangle from Vertices

Coordinate Geometry · Class X

Place three vertices and compute the signed and unsigned triangle area.

0
type a value
-1010
0
type a value
-1010
6
type a value
-1010
0
type a value
-1010
Live values
  • Signed area (C at (0,4))12
  • Absolute area12
  • Are A, B, (0,4) collinear?0
xy
  • Area as A.x slides

Formulas in this lab

  • Signed area (C at (0,4))
    12∣xA(yB−yC)+xB(yC−yA)+xC(yA−yB)∣\tfrac{1}{2}|x_A(y_B-y_C)+x_B(y_C-y_A)+x_C(y_A-y_B)|
  • Absolute area
    ∣signed area∣|\text{signed area}|
  • Are A, B, (0,4) collinear?
    area=0?\text{area} = 0?
Tip: Area is a V-shape in A.x; the zero of the V marks collinearity (no triangle).

Frequently asked questions

▶How do we find a triangle's area from vertices?

For vertices $A(x_1, y_1)$, $B(x_2, y_2)$, $C(x_3, y_3)$, the area is $\frac{1}{2}|x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$. If the expression inside the bars is zero, the three points are collinear.

▶How do I use this lab?

Place A, B, C with sliders. The lab computes signed and unsigned area instantly and warns if the three points are collinear. Try $A(0,0)$, $B(4,0)$, $C(0,3)$ to see area 6 square units.

▶Common mistake on this topic

Students forget to take the absolute value, ending up with a negative area. Also, they sometimes write $y_3 - y_2$ instead of $y_2 - y_3$ in the first term. Following the cyclic pattern $1 \to 2 \to 3 \to 1$ helps.

▶Where do we use this?

Land-revenue officials compute plot areas from corner coordinates without going to the field. In mapping apps, when you draw a triangular region for delivery, the same formula calculates its area.