7.3 Areas of Polygons
In coordinate geometry, the area of any polygon on the Cartesian plane can be determined directly if the coordinates of its vertices are known. Instead of calculating perpendicular heights and base lengths using Euclidean geometry, we can use the shoelace algorithm or the box method.
Learning Objectives
- Derive the formula for the area of a triangle when the coordinates of each vertex are known.
- Determine the area of triangles using the shoelace formula and the box method.
- Apply the collinearity condition (\(\text{Area} = 0\)) to show or find unknown values of collinear points.
- Determine the area of quadrilaterals and general \(n\)-sided polygons using the extended shoelace algorithm.
- Solve geometric and real-world problems involving areas of polygons, unknown coordinates, and area ratios.
The Shoelace Algorithm Pattern and Directional Signs
Illustrate how coordinates are arranged in an array, how criss-cross arrows define the downward (\(\searrow\) positive) and upward (\(\nearrow\) negative) products, and how the first vertex is repeated.
Array arrangement
\[ \begin{matrix} x_1 & x_2 & x_3 & x_1 \\ y_1 & y_2 & y_3 & y_1 \end{matrix} \]
Directional signs
Downward diagonal products (\(\searrow\)) are assigned a positive sign.
Upward diagonal products (\(\nearrow\)) are assigned a negative sign.
Anticlockwise order
Yields a positive value directly. The absolute value ensures the final area is positive.
Clockwise order
Yields a negative numerical value. Taking the absolute value gives the correct positive area.
Short Explanation
The shoelace algorithm computes the area of any polygon directly from its coordinates.
1. Derivation of the Area of a Triangle
Consider \(\Delta ABC\) with vertices \(A(x_1, y_1)\), \(B(x_2, y_2)\), and \(C(x_3, y_3)\). By projecting vertical lines down to a baseline: \[ \text{Area of } \Delta ABC = \text{Area of } \Delta ACD + \text{Area of trapezium } BCDE - \text{Area of } \Delta ABE \] Expanding and simplifying yields: \[ \text{Area of } \Delta ABC = \frac{1}{2}(x_1 y_2 + x_2 y_3 + x_3 y_1 - x_2 y_1 - x_3 y_2 - x_1 y_3) \]
2. The Shoelace Formula for Triangles
Written in vertical array notation: \[ \text{Area of } \Delta ABC = \frac{1}{2} \left| \begin{matrix} x_1 & x_2 & x_3 & x_1 \\ y_1 & y_2 & y_3 & y_1 \end{matrix} \right| \] Combined absolute value form: \[ \text{Area} = \frac{1}{2} |(x_1 y_2 + x_2 y_3 + x_3 y_1) - (x_2 y_1 + x_3 y_2 + x_1 y_3)| \]
3. Alternative: The Box Method
Step 1: Draw an enclosing bounding rectangle whose sides are parallel to the coordinate axes and pass through each vertex of the triangle.
Step 2: Calculate the area of the rectangle: \(\text{Length} \times \text{Width}\).
Step 3: Calculate the areas of the outer right-angled triangles (\(I, II, III\)) formed between the target triangle and the box boundary: \(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\).
Step 4: Subtract the areas of the outer triangles from the rectangle area:
\[
\text{Area of } \Delta ABC = \text{Area of rectangle} - (\text{Area } I + \text{Area } II + \text{Area } III)
\]
4. Collinear Points
Three points \(A(x_1, y_1)\), \(B(x_2, y_2)\), and \(C(x_3, y_3)\) are collinear if and only if the area of the triangle formed by them is zero: \[ \frac{1}{2} |(x_1 y_2 + x_2 y_3 + x_3 y_1) - (x_2 y_1 + x_3 y_2 + x_1 y_3)| = 0 \]
5. Area of Quadrilaterals
A quadrilateral \(ABCD\) can be partitioned into two triangles: \(\Delta ABC + \Delta ACD\). Expanding gives the shoelace formula for four vertices arranged in order: \[ \text{Area of quadrilateral } ABCD = \frac{1}{2} \left| \begin{matrix} x_1 & x_2 & x_3 & x_4 & x_1 \\ y_1 & y_2 & y_3 & y_4 & y_1 \end{matrix} \right| \] \[ = \frac{1}{2} |(x_1 y_2 + x_2 y_3 + x_3 y_4 + x_4 y_1) - (x_2 y_1 + x_3 y_2 + x_4 y_3 + x_1 y_4)| \]
6. Generalisation to Polygons with \(n\) Sides
For any polygon with \(n\) vertices \(A(x_1, y_1), B(x_2, y_2), \dots, N(x_n, y_n)\) arranged consecutively around the perimeter: \[ \text{Area of polygon} = \frac{1}{2} \left| \begin{matrix} x_1 & x_2 & x_3 & \dots & x_n & x_1 \\ y_1 & y_2 & y_3 & \dots & y_n & y_1 \end{matrix} \right| \] \[ = \frac{1}{2} |(\text{sum of all } \searrow \text{ products}) - (\text{sum of all } \nearrow \text{ products})| \]
7. Solving Problems Involving Unknown Coordinates
When the numerical area is given and one coordinate is an unknown parameter (e.g., \(k\)): \[ \frac{1}{2}(\text{expression}) = +\text{Area} \quad \text{or} \quad \frac{1}{2}(\text{expression}) = -\text{Area} \] This typically produces two valid values of \(k\), corresponding to two different geometric positions that enclose the same area.
General shoelace formula
\(\text{Area} = \dfrac{1}{2} |(x_1 y_2 + x_2 y_3 + \dots + x_n y_1) - (x_2 y_1 + x_3 y_2 + \dots + x_1 y_n)|\)List vertices in sequential order around the perimeter and repeat the first vertex at the end.
Try to Answer First
Answer in your mind, then press "Check Answer".
Find the area of \(\Delta ABC\) with vertices \(A(3, 1)\), \(B(6, 4)\), and \(C(-4, 2)\).
Upward products \((\nearrow) = (1)(6) + (4)(-4) + (2)(3) = 6 - 16 + 6 = -4\).
\(\text{Area} = \frac{1}{2} |20 - (-4)| = \frac{1}{2} |24| = 12 \text{ units}^2\).
Show that the points \(P(8, 4)\), \(Q(2, 1)\), and \(R(-2, -1)\) are collinear.
Upward products \((\nearrow) = (4)(2) + (1)(-2) + (-1)(8) = 8 - 2 - 8 = -2\).
\(\text{Area} = \frac{1}{2} |(-2) - (-2)| = \frac{1}{2} |0| = 0\).
Since the area of \(\Delta PQR = 0\), the points \(P\), \(Q\), and \(R\) are collinear.
The vertices of \(\Delta ABC\) are \(A(8, 5)\), \(B(-2, -3)\), and \(C(k, -1)\). Given that the area of \(\Delta ABC\) is \(18\text{ units}^2\), find the possible values of \(k\).
Upward products \(= (5)(-2) + (-3)(k) + (-1)(8) = -10 - 3k - 8 = -3k - 18\).
Difference \(= (5k - 22) - (-3k - 18) = 8k - 4\).
\(\frac{1}{2} |8k - 4| = 18 \implies |8k - 4| = 36\).
Case 1: \(8k - 4 = 36 \implies 8k = 40 \implies k = 5\).
Case 2: \(8k - 4 = -36 \implies 8k = -32 \implies k = -4\).
Thus, the possible values of \(k\) are \(-4\) and \(5\).
Common Mistakes
- Forgetting to repeat the first vertex \((x_1, y_1)\) at the very end of the array, leading to incomplete products.
- Arranging vertices across diagonals or at random rather than following consecutive order along the polygon's perimeter.
- Omitting negative signs during multiplication (e.g., \((-2) \times (-3) = +6\), not \(-6\)).
- Forgetting to apply both positive and negative cases (\(\pm \text{Area}\)) when solving for an unknown coordinate from a given area.
- Forgetting to divide the final difference by \(2\).
Concept Misunderstandings
The shoelace formula only works if vertices are taken in anticlockwise order.
Vertices can be taken in clockwise order as well. Clockwise order yields a negative value inside the brackets, but taking the absolute value gives the correct positive area.
The shoelace formula only applies to triangles.
The formula applies to any polygon with \(n\) sides (triangles, quadrilaterals, pentagons, hexagons, etc.) as long as vertices are taken in sequential order around the perimeter.
Area can be negative if coordinates have negative values.
Area is a scalar quantity measuring size/magnitude, so it is strictly non-negative. The vertical bars denote absolute value.
Summary
- Shoelace Formula: \(\text{Area} = \frac{1}{2} |(x_1 y_2 + x_2 y_3 + \dots + x_n y_1) - (x_2 y_1 + x_3 y_2 + \dots + x_1 y_n)|\).
- Order Matters: Always arrange vertices sequentially around the boundary. Loop back to the first vertex.
- Collinearity Test: Three points are collinear if and only if \(\text{Area} = 0\).
- Solving for Unknowns: An equation \(\frac{1}{2}|\text{expression}| = A\) must be solved as \(\text{expression} = \pm 2A\), typically yielding two valid answers.
- Box Method: Provides a geometric alternative by subtracting surrounding right-angled triangles from a bounding rectangle.
Short Activity
Objective: Identify collinearity, clockwise orientation, diagonal signs, and key terms in areas of polygons.
A. Objective Quiz
1 What does it indicate about three distinct points \(A\), \(B\), and \(C\) if the value of \(\frac{1}{2} \left| \begin{matrix} x_1 & x_2 & x_3 & x_1 \\ y_1 & y_2 & y_3 & y_1 \end{matrix} \right|\) is equal to \(0\)?
2 When using the shoelace algorithm, what is the result if the vertices of a polygon are arranged in a clockwise direction?
B. Fill in the Blanks
3 In the shoelace formula, the diagonal products in the down-right direction (\(\searrow\)) are assigned a ________ sign.
4 Area is a ________ quantity because it has magnitude only and does not involve direction.
C. Matching / Drag and Drop
Drag each term to the correct meaning. On a phone, tap a term first, then tap the matching box.
