<p>When <strong>3 or more points</strong> are joined to form a polygon on the Cartesian plane, its area can be found using the <strong>Shoelace Method</strong>.<br>Suppose a polygon has vertices \( (x_1, y_1), (x_2, y_2), (x_3, y_3), \dots, (x_n, y_n) \)<br>The area is given by:</p>
<p style="text-align: center">Area \( = \frac{1}{2} \left| \begin{matrix} x_1 & x_ 2 & x_3 & \dots & x_n & x_1\\ y_1 & y_2 & y_3 & \dots & y_n &y_1 \end{matrix} \right| \)<br>Area \( = \frac{1}{2} \left| [x_1y_2 + x_2y_3 + x_3y_4 + \dots + x_ny_1] - [y_1x_2 + y_2x_3 +y_3x_4 + \dots + y_nx_1] \right| \)<br>where the polygon has&nbsp;\(n\) vertices</p>
<p><strong>How to use the Shoelace Method:</strong></p>
<ol class="num-list"><li data-list-item-id="item-1782729775284-4snzss57y" data-list-item-path="item-1782729775284-4snzss57y"><div class="list-item-paragraph">Write the coordinates in the order that the vertices are connected, either <strong>clockwise</strong> or <strong>anti-clockwise</strong>.</div></li><li data-list-item-id="item-1782729775865-nzwzdwrmd" data-list-item-path="item-1782729775865-nzwzdwrmd"><div class="list-item-paragraph"><strong>Repeat</strong> the first point at the end of the table.</div></li><li data-list-item-id="item-1782729776432-3lcdz73e6" data-list-item-path="item-1782729776432-3lcdz73e6"><div class="list-item-paragraph">Find the first sum by multiplying each \(x\)-coordinate by the \(y\)-coordinate in the next row (downwards to the right).</div></li><li data-list-item-id="item-1782729776951-ialxowfmt" data-list-item-path="item-1782729776951-ialxowfmt"><div class="list-item-paragraph">Find the second sum by multiplying each \(y\)-coordinate by the next \(x\)-coordinate (upwards to the right).</div></li><li data-list-item-id="item-1782729853393-j1cj8z0k2" data-list-item-path="item-1782729853393-j1cj8z0k2"><div class="list-item-paragraph">Subtract the second sum from the first sum.</div></li><li data-list-item-id="item-1782729777496-jnnjrn4bs" data-list-item-path="item-1782729777496-jnnjrn4bs"><div class="list-item-paragraph">Take the <strong>absolute value</strong> (ignore any negative sign), then divide the result by \(2\).</div></li></ol>
<p><strong>Important Notes</strong></p>
<ul class="bullet-list"><li data-list-item-id="item-1782729997257-2tq9lhwhr" data-list-item-path="item-1782729997257-2tq9lhwhr"><div class="list-item-paragraph">The vertices must be listed <strong>in the order they are connected</strong> around the polygon.</div></li><li data-list-item-id="item-1782729997684-cuxph478x" data-list-item-path="item-1782729997684-cuxph478x"><div class="list-item-paragraph">The order can be <strong>clockwise</strong> or <strong>anti-clockwise</strong>, but do not mix the order.</div></li><li data-list-item-id="item-1782729998101-w5bbi3dl8" data-list-item-path="item-1782729998101-w5bbi3dl8"><div class="list-item-paragraph">Always <strong>repeat the first vertex</strong> at the end before carrying out the multiplications.</div></li><li data-list-item-id="item-1782729998548-oww9bouh5" data-list-item-path="item-1782729998548-oww9bouh5"><div class="list-item-paragraph">Use the <strong>absolute value</strong> to ensure the area is always positive.</div></li></ul>

When **3 or more points** are joined to form a polygon on the Cartesian plane, its area can be found using the **Shoelace Method**.
Suppose a polygon has vertices \( (x_1, y_1), (x_2, y_2), (x_3, y_3), \dots, (x_n, y_n) \)
The area is given by:

Area \( = \frac{1}{2} \left| \begin{matrix} x_1 & x_ 2 & x_3 & \dots & x_n & x_1\\ y_1 & y_2 & y_3 & \dots & y_n &y_1 \end{matrix} \right| \)
Area \( = \frac{1}{2} \left| [x_1y_2 + x_2y_3 + x_3y_4 + \dots + x_ny_1] - [y_1x_2 + y_2x_3 +y_3x_4 + \dots + y_nx_1] \right| \)
where the polygon has \(n\) vertices

Write the coordinates in the order that the vertices are connected, either **clockwise** or **anti-clockwise**.

**Repeat** the first point at the end of the table.

Find the first sum by multiplying each \(x\)-coordinate by the \(y\)-coordinate in the next row (downwards to the right).

Find the second sum by multiplying each \(y\)-coordinate by the next \(x\)-coordinate (upwards to the right).

Subtract the second sum from the first sum.

Take the **absolute value** (ignore any negative sign), then divide the result by \(2\).

The vertices must be listed **in the order they are connected** around the polygon.

The order can be **clockwise** or **anti-clockwise**, but do not mix the order.

Always **repeat the first vertex** at the end before carrying out the multiplications.

Use the **absolute value** to ensure the area is always positive.

Area of Rectilinear Polygons

Area of Rectilinear Polygons