How to Find the Area of a Polygon Using Coordinates?
The concept of area of polygon plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding polygon area formulas, differences between regular and irregular polygons, and efficient problem-solving methods is essential for students and useful in many fields.
What Is Area of Polygon?
A polygon is a two-dimensional closed shape formed by straight line segments. The area of a polygon refers to the region or space enclosed by its sides. This concept is used while working with triangles, rectangles, pentagons, hexagons, and any many-sided flat shape. You’ll find this concept applied in geometry, coordinate geometry, and even in dividing land or creating models in real life.
Key Formula for Area of Polygon
Here’s the standard formula for a regular n-sided polygon:
\[
\text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem}
\]
Or,
\[
\text{Area} = \frac{n \times s^2}{4 \times \tan( \frac{\pi}{n} )}
\]
Where:
s = length of each side
For polygons with vertices on the coordinate plane: \[ \text{Area} = \frac{1}{2} \left| \sum_{i=1}^{n} (x_i y_{i+1} - x_{i+1} y_i) \right| \] This is known as the Shoelace Formula.
Step-by-Step Illustration
- Start with a regular pentagon with side length 6 cm.
Number of sides, n = 5,
Length of each side, s = 6 cm. - Find the apothem (a):
Use \( a = \frac{s}{2\tan(\pi/n)} \)
\( a = \frac{6}{2\tan(\pi/5)} \approx 4.12 \) cm. - Find the perimeter:
Perimeter = n × s = 5 × 6 = 30 cm. - Apply the area formula:
Area = ½ × Perimeter × Apothem
Area = ½ × 30 × 4.12 = 61.8 cm2
Area of Polygon in the Coordinate Plane
For polygons given by points (x, y), use the Shoelace formula. List coordinates in order and apply these steps:
1. Multiply each x-coordinate by the next y-coordinate (wrap around at the end).2. Multiply each y-coordinate by the next x-coordinate.
3. Find the sum for both.
4. Area = ½ |sum1 − sum2|.
Example: For triangle with A(1,2), B(4,5), C(7,8):
1. (1×5 + 4×8 + 7×2) = 5 + 32 + 14 = 51
2. (2×4 + 5×7 + 8×1) = 8 + 35 + 8 = 51
3. Area = ½ |51 − 51| = 0 (which means the points are collinear; try with non-collinear points for real area value!)
Area of Irregular Polygons
For shapes that are not regular, divide the shape into known figures (like triangles, rectangles, or trapeziums), calculate the area of each, and then add them up. This method is especially useful in practical situations like plotting land or solving complex competitive exam questions.
Quick Area Reference Table
| Polygon Type | Formula |
|---|---|
| Triangle | ½ × base × height |
| Square | side × side |
| Rectangle | length × width |
| Regular Pentagon | (5 × s²) / (4 × tan(π/5)) |
| Regular Hexagon | (3√3 / 2) × s² |
| n-sided Regular Polygon | (n × s²) / [4 × tan(π/n)] |
| Polygon with coordinates | ½ × |(sum of xiyi+1 − xi+1yi)| |
Speed Trick or Vedic Shortcut
When a polygon has all sides and apothem given, use the “½ × Perimeter × Apothem” shortcut instead of calculating area triangle by triangle. This is fast for competitive and board exams. If the vertices are in coordinates, immediately set up the Shoelace steps instead of plotting or breaking into triangles — it saves time.
Try These Yourself
- Find the area of a regular hexagon with side 10 cm.
- Given vertices (0,0), (4,0), (4,3), (0,3), find the area of the quadrilateral.
- Divide an L-shaped figure into rectangles and triangles, then find its area.
- What is the area of a pentagon with side 7 cm (use tan 36° ≈ 0.7265)?
Frequent Errors and Misunderstandings
- Mixing up ‘perimeter’ and ‘area’ formulas.
- Applying a regular polygon formula to an irregular shape.
- Swapping x and y coordinates in the Shoelace formula.
- Not closing the loop (forgetting to connect last vertex to the first).
- Missing or wrong units in the final answer.
Cross-Disciplinary Usage
Area of polygon is not only useful in Maths but also plays an important role in Physics (for surface calculations), Computer Science (graphics, GIS), and logical reasoning. Students preparing for JEE or Olympiads will encounter polygon area questions in geometry, mensuration, and coordinate geometry sections.
Relation to Other Concepts
The idea of area of polygon connects with concepts like Area of Parallelogram and Area of a Triangle. Many irregular polygons can be divided into these basic shapes for easier computation. Also, perimeter and area are often compared for surface analysis.
Classroom Tip
A practical way to remember area formulas for polygons is by drawing the shape and labeling all sides and center lines (like apothem). For polygons on a coordinate plane, always write the vertex list in order and finish by connecting the last point to the first. Vedantu’s stepwise live explanations help reinforce these process-based tips.
We explored area of polygon—from definition, formula, coordinate shortcuts, and common errors, to real-life uses and practice questions. Continue practicing with Vedantu to build confidence and mastery in all related geometry concepts.
Explore more: Perimeter of a Polygon | Area of Parallelogram | Area of a Triangle | Types of Polygon
FAQs on Area of Polygons – Formulas, Steps & Solved Examples
1. What is a polygon and how is its area defined?
A polygon is a two-dimensional, closed shape made of straight line segments. The area of a polygon is the total amount of surface enclosed within its boundary. It is measured in square units, such as square centimetres (cm²) or square metres (m²), and represents the space the polygon occupies in a plane.
2. What is the fundamental difference between calculating the area of a regular versus an irregular polygon?
The key difference lies in the uniformity of the shapes.
- A regular polygon has all sides and all interior angles equal. This uniformity allows us to use direct formulas, such as those involving the apothem or side length.
- An irregular polygon has sides and/or angles of different measures. There is no single direct formula; instead, we must use methods like decomposition (dividing it into simpler shapes like triangles) or the Shoelace formula if vertex coordinates are known.
3. How is the area of a regular polygon calculated using its apothem?
The area of a regular polygon can be found using the formula: Area = (1/2) × Perimeter × Apothem. The apothem is the perpendicular distance from the center of the polygon to the midpoint of any side. This formula works because it effectively breaks the polygon into a series of identical triangles, with the apothem as their height.
4. What is the general formula to find the area of a regular n-sided polygon if only the side length is known?
When the apothem is not given, you can calculate the area of a regular polygon with 'n' sides and a side length 's' using the formula: Area = (n × s²) / [4 × tan(π/n)]. Here, 'n' is the number of sides, 's' is the length of a side, and the angle is measured in radians.
5. How do you find the area of an irregular polygon by dividing it into simpler shapes?
This method, known as decomposition, involves strategically drawing lines to break down the complex irregular polygon into familiar, simpler shapes for which area formulas are known. Common shapes to use are:
- Triangles
- Rectangles
- Trapezoids
6. What is the Shoelace Formula and when is it used to find a polygon's area?
The Shoelace Formula (or Surveyor's Formula) is a powerful method used in coordinate geometry to find the area of any simple polygon when the coordinates of its vertices (x, y) are known. You list the coordinates in counterclockwise or clockwise order and apply the formula: Area = 0.5 × |(x₁y₂ + x₂y₃ + ... + xₙy₁) - (y₁x₂ + y₂x₃ + ... + yₙx₁)|. It is especially useful for irregular polygons on a Cartesian plane.
7. How is the formula for the area of a regular hexagon derived from its side length?
The derivation is based on a simple insight: a regular hexagon can be divided into six congruent equilateral triangles, with each side of the triangles being equal to the side length 's' of the hexagon. The area of one equilateral triangle is given by the formula (√3/4) × s². Since there are six such triangles, the total area of the hexagon is 6 × (√3/4) × s², which simplifies to (3√3/2) × s².
8. Can the method for finding the area of a convex polygon be applied to a concave polygon? Explain why or why not.
Yes, the primary methods work for both. The decomposition method (dividing into triangles) is applicable, though for a concave polygon, you must be careful how you triangulate the interior 'dent'. More reliably, the Shoelace Formula works perfectly for both convex and concave polygons without any change in method. Its mathematical structure of summing signed areas automatically handles the 'dented' regions as long as the vertices are listed in consecutive order.
9. Beyond geometry class, what are some real-world applications where calculating the area of polygons is essential?
Calculating polygon areas is a fundamental skill used in many professional fields, including:
- Architecture and Construction: To determine the floor area of a building, calculate material requirements like tiles or paint, and design layouts.
- Land Surveying: To measure the precise area of a plot of land, which is often an irregular polygon, for legal and commercial purposes.
- Computer Graphics and Gaming: To render 2D shapes, calculate the space an object occupies on screen, and define collision boundaries in video games.
- Urban Planning: To calculate the area of parks, residential zones, or industrial districts on a city map.
10. Why does the Shoelace Formula work for both regular and irregular polygons?
The Shoelace Formula's power comes from its basis in coordinate geometry, not the polygon's properties like equal sides or angles. It works by calculating the signed area of trapezoids formed between each polygon edge and a coordinate axis. The formula systematically adds and subtracts these areas. Because this process depends only on the sequential coordinates of the vertices, it is indifferent to whether the side lengths or angles are uniform (regular) or varied (irregular). It calculates the total enclosed area based purely on the boundary points.
11. What is a common misconception when choosing a formula for a polygon's area?
A very common misconception is applying a formula meant for a regular polygon to an irregular polygon. For example, using the apothem-based formula (Area = ½ × Perimeter × Apothem) for a polygon where sides are not equal is incorrect, as an irregular polygon does not have a single, well-defined apothem. One must first identify the polygon type: if it is regular, use the specific formulas; if it is irregular, use general methods like decomposition or the Shoelace formula.
