Area of Triangle and Quadrilateral

Area of Triangle

Let A(x₁,y₁), B(x₂.y₂) and C (x_3.y₃) be the vertices of a triangle. Draw \(\perp\)AL, BM and CN from the vertices A, B, and C respectively to the X-axis.

Then,
OL = x₁ OM = x₂ ON =x₃
AL = y₁ BM = y₂ CN = y₃
Here, LN = ON - OL = x₃-x₁
NM = OM - ON = x₂-x₃ and
LM = OM -OL = x₂-x₁
Now, Area ofΔABC is equal to
Area of trapezium ALNC + Area of trapezium CNMB - Area of trapezium ALMB.
= ½LN(AL+CN) + ½NM(CN+BM) - ½ LM(AL+BM)
= ½(x_3-x₁)(y₁+y₃)+½(x₂-x₃)(y₃+y₂)-½(x₂-x₁)(y₁+y₂)
= ½(x₃y₁+x₃y₃-x₁y₁-x₁y₃+x₂y₃+x₂y₂-x₃y₃-x₃y₂-x₂y₁-x₂y₂+x₁y₁+x₁y₂)
= ½(x₁y₂-x₂y₁+x₂y₃-x₃y₂+x₃y₁-x₁y₃)

Area of a Quadrilateral

Let A (x₁,y₁), B(x₂,y₂), C(x_3,y₃) and D (x₄,y₄)be the vertices of quadrilateral ABCD. Join AC. Then the diagonal AC divides the quadrilateral into two triangles ABC and ACD.
Now,
Area of quadrilateral ABCD = Area of \(\triangle\)ABC + Area of \(\triangle\)ACD.
\(\square\)ABCD = ½\(\begin{vmatrix}x_1&x_2&x_3&x_1\\y_1&y_2&y_3&y_1 \end{vmatrix}\) + ½ \(\begin{vmatrix}x_1&x_3&x_4&x_1\\y_1&y_3&y_4&y_1 \end{vmatrix}\)
= ½ (x₁y₂-x₂y₁+x₂y₃-x₃y₂+x₃y₁-x₁y₃) + ½ (x₁y₃-x₃y₁+x₃y₄-x₄y₃+x₄y₁-x₁y₄)
= ½ (x₁y₂-x₂y₁+x₂y₃-x₃y₂+x₃y₄-x₄y₃+x₄y₁-x₁y₄)

Note: To find the area of a quadrilateral by using this formula, the vertices of quadrilateral should be taken in order. So it is better to plot the points roughty before applying the formula. Otherwise the result may be wrong.