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 the formula for the area of a regular polygon?
The area of a regular polygon can be calculated using the formula: Area = (1/2) × Perimeter × Apothem. The apothem is the distance from the center of the polygon to the midpoint of any side. Alternatively, for a regular n-sided polygon with side length s, the area can be calculated as: Area = (n × s²) / [4 × tan(π/n)].
2. How can I calculate the area of a polygon with given vertices?
To find the area of a polygon given its vertices' coordinates, use the Shoelace Formula (also known as the Surveyor's Formula). This method involves a systematic process of multiplying and summing coordinates. The formula is: Area = 0.5 × |(x₁y₂ + x₂y₃ + ... + xₙy₁) - (y₁x₂ + y₂x₃ + ... + yₙx₁)|, where (xᵢ, yᵢ) are the coordinates of the vertices. Remember to list the vertices in order, either clockwise or counterclockwise.
3. How is the area of an irregular polygon found?
The area of an irregular polygon is calculated by dividing it into smaller, simpler shapes like triangles, rectangles, or trapezoids, for which area formulas are readily available. Calculate the area of each smaller shape individually and then sum the areas to find the total area of the irregular polygon. Using coordinate geometry, the Shoelace Formula can also be applied to irregular polygons.
4. What is an apothem in polygon area calculation?
In the context of calculating the area of a regular polygon, the apothem is the perpendicular distance from the center of the polygon to the midpoint of any side. It's a crucial element in the formula Area = (1/2) × Perimeter × Apothem.
5. Can I use an online area of polygon calculator for exams?
While online calculators can be helpful for verification, it's generally recommended to understand and apply the area formulas yourself during exams. Calculators might not be permitted, and demonstrating your understanding of the concepts is crucial for good marks. However, using a calculator to check your answers after the exam is a useful way to verify your work.
6. What is the area of a polygon with coordinates (1,1), (4,1), (4,3), (1,3)?
This polygon is a rectangle. Using the formula for the area of a rectangle, Area = length × width, we find: Area = (4-1) × (3-1) = 3 × 2 = 6 square units. Alternatively, using the Shoelace Formula will also yield the same result.
7. How do I find the area of a hexagon given its side length?
The area of a regular hexagon can be calculated using its side length (s). One formula is: Area = (3√3/2) × s². This formula is derived by dividing the hexagon into six equilateral triangles.
8. Explain the difference between a regular and irregular polygon.
A regular polygon has all sides of equal length and all interior angles of equal measure. An irregular polygon has sides and/or angles of varying lengths and measures.
9. Can the Shoelace Formula be used for concave polygons?
Yes, the Shoelace Formula works for both convex and concave polygons. The key is to list the vertices in consecutive order (either clockwise or counter-clockwise).
10. What are some common mistakes to avoid when calculating polygon areas?
Common mistakes include: using the wrong formula for the type of polygon; incorrect measurement of sides or apothem; errors in applying the Shoelace Formula (especially sign errors); forgetting to square side lengths when appropriate; and not using consistent units throughout calculations.
11. How can I visually represent polygon area calculations?
Visual aids are very helpful! Use diagrams to divide irregular polygons into smaller shapes; clearly label sides, angles, and coordinates; draw the apothem for regular polygons. Graphing the vertices and connecting them is essential when using the Shoelace method. Use different colors to represent different sections of a complex polygon.
12. Are there real-world applications for calculating polygon areas?
Yes! Calculating polygon areas is used extensively in: land surveying (measuring property boundaries); architecture and construction (calculating floor space, material needs); computer graphics (rendering 2D shapes); and even in some aspects of game design (creating and managing in-game environments).
