Calculating the Area of a Triangle from Vertices

Introduction

• The area of a triangle can be calculated using the three vertices.

Formula

• Formula: ( \text{Area} = \frac{1}{2} \left| \text{det}(A) \right| )
  • Where ( A ) is a 3x3 matrix.

Defining Matrix A

• Matrix A is constructed as follows:
  • First Column: ( x_1, x_2, x_3 )
  • Second Column: ( y_1, y_2, y_3 )
  • Third Column: Three ones.

Example 1

• Given vertices: (1, 1), (4, 1), (4, 5)
• Matrix A: [ \begin{bmatrix} 1 & 1 & 1 \ 4 & 1 & 1 \ 4 & 5 & 1 \end{bmatrix} ]

Evaluating the Determinant

• Calculate determinant:
  1. From top left:
     • ( 1 \times 1 \times 1 = 1 )
     • ( 1 \times 1 \times 4 = 4 )
     • ( 1 \times 4 \times 5 = 20 )
  2. From bottom left:
     • ( 4 \times 1 \times 1 = 4 )
     • ( 5 \times 1 \times 1 = 5 )
     • ( 1 \times 4 \times 1 = 4 )
• Add values:
  • Top: ( 1 + 4 + 20 = 25 )
  • Bottom: ( 4 + 5 + 4 = 13 )
• ( ext{Det}(A) = 25 - 13 = 12 )

Area Calculation

• Area = ( \frac{1}{2} \times 12 = 6 ) square units
• Verification
  • Right triangle with base = 3, height = 4.
  • Area = ( \frac{1}{2} \times 3 \times 4 = 6 )

Example 2

• New vertices: (2, 3), (5, 7), (10, -5)
• Matrix A: [ \begin{bmatrix} 2 & 3 & 1 \ 5 & 7 & 1 \ 10 & -5 & 1 \end{bmatrix} ]

Evaluating the Determinant

• Calculate determinant:
  1. Multiply from top left:
     • ( 2 \times 7 \times 1 = 14 )
     • ( 3 \times 1 \times 10 = 30 )
     • ( 1 \times 5 \times -5 = -25 )
  2. Multiply from bottom left:
     • ( 10 \times 7 \times 1 = 70 )
     • ( -5 \times 1 \times 2 = -10 )
     • ( 1 \times 5 \times 3 = 15 )
• Add values:
  • Top: ( 14 + 30 - 25 = 19 )
  • Bottom: ( 70 - 10 + 15 = 75 )
• ( ext{Det}(A) = 19 - 75 = -56 )

Area Calculation

• Area = ( \frac{1}{2} \times | -56 | = 28 ) square units

Verification by Graphing

• Plotting points leads to a scaling triangle.

Alternative Method: Heron's Formula

1. Calculate side lengths.
2. Determine semi-perimeter ( S ).
3. Calculate area using: [ \text{Area} = \sqrt{S(S-a)(S-b)(S-c)} ]

Conclusion

• The easiest way to calculate the area of a triangle from vertices is through the determinant of the 3x3 matrix and dividing by 2.