Regular Polygon Area Calculator
Calculate the area of a regular polygon using the number of sides and either the side length, apothem, or radius.
Results:
Perimeter (P): –
Interior Angle: –
Exterior Angle: –
Calculated Side Length (s): –
Calculated Apothem (a): –
Calculated Radius (r): –
Area vs. Number of Sides
Area Table
| Sides (n) | Area (given dimension) | Area (if side was 10) |
|---|---|---|
| Enter values to see table data. | ||
What is the Regular Polygon Area?
The Regular Polygon Area is the measure of the two-dimensional space enclosed within the boundaries of a regular polygon. A regular polygon is a polygon that is both equiangular (all angles are equal) and equilateral (all sides have the same length). Calculating the Regular Polygon Area is a fundamental concept in geometry, used in various fields like architecture, engineering, design, and land surveying.
Anyone needing to find the surface area of a regular polygonal shape should use a Regular Polygon Area calculator or formula. This includes students learning geometry, architects designing structures with regular polygonal bases, or engineers calculating material requirements.
Common misconceptions include confusing the area with the perimeter (the distance around the polygon) or thinking that all polygons with the same number of sides have the same area (which is only true if they are regular and have the same side length, apothem, or radius).
Regular Polygon Area Formula and Mathematical Explanation
The formula for the Regular Polygon Area depends on the information you have about the polygon: the number of sides (n), the length of one side (s), the apothem (a – the distance from the center to the midpoint of a side), or the radius (r – the distance from the center to a vertex).
1. Given Number of Sides (n) and Side Length (s):
The most common formula when the side length is known is:
Area = (n * s²) / (4 * tan(π / n))
Where: n is the number of sides, s is the side length, and tan is the tangent function (π/n is in radians).
This formula is derived by dividing the polygon into n congruent isosceles triangles, each with a base s, and finding the area of one triangle using its height (the apothem) and then multiplying by n.
2. Given Number of Sides (n) and Apothem (a):
Area = n * a² * tan(π / n)
Alternatively, if you first find the perimeter P = n * s, and s = 2 * a * tan(π/n), then:
Area = 0.5 * P * a = 0.5 * (n * 2 * a * tan(π/n)) * a = n * a² * tan(π/n)
3. Given Number of Sides (n) and Radius (r):
Area = 0.5 * n * r² * sin(2π / n)
This formula is also derived by dividing the polygon into n triangles, using the formula for the area of a triangle given two sides and the included angle (0.5 * r * r * sin(2π/n)).
Intermediate values often calculated are:
- Perimeter (P) = n * s
- Interior Angle = (n-2) * 180 / n degrees
- Exterior Angle = 360 / n degrees
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| n | Number of sides | – | ≥ 3 (integer) |
| s | Side length | Length (e.g., m, cm, ft) | > 0 |
| a | Apothem | Length (e.g., m, cm, ft) | > 0 |
| r | Radius (or circumradius) | Length (e.g., m, cm, ft) | > 0 |
| Area | Area of the polygon | Area (e.g., m², cm², ft²) | > 0 |
| P | Perimeter | Length | > 0 |
Calculating the Regular Polygon Area is crucial for various applications.
Practical Examples (Real-World Use Cases)
Example 1: Tiling a Floor
Imagine you are tiling a floor with regular hexagonal tiles, each with a side length (s) of 15 cm. You need to find the area of one tile to estimate the total number of tiles needed.
- Number of sides (n) = 6
- Side length (s) = 15 cm
- Using the formula: Area = (6 * 15²) / (4 * tan(π / 6)) = (6 * 225) / (4 * tan(30°)) = 1350 / (4 * 0.57735) ≈ 1350 / 2.3094 ≈ 584.56 cm²
- The Regular Polygon Area of one tile is approximately 584.56 cm².
Example 2: Designing a Gazebo Base
An architect is designing a gazebo with a regular octagonal base. The distance from the center to any corner (radius, r) is 3 meters.
- Number of sides (n) = 8
- Radius (r) = 3 m
- Using the formula: Area = 0.5 * 8 * 3² * sin(2π / 8) = 4 * 9 * sin(π/4) = 36 * sin(45°) ≈ 36 * 0.7071 ≈ 25.46 m²
- The Regular Polygon Area of the gazebo base is approximately 25.46 m².
These examples show how the Regular Polygon Area calculation is applied. Check out our geometry calculators for more tools.
How to Use This Regular Polygon Area Calculator
- Enter the Number of Sides (n): Input the total number of sides your regular polygon has (e.g., 3 for a triangle, 5 for a pentagon, 8 for an octagon). It must be 3 or more.
- Select Known Dimension: Choose whether you know the 'Side Length', 'Apothem', or 'Radius' of the polygon by clicking the corresponding radio button.
- Enter the Known Value: Based on your selection in step 2, enter the value for the side length (s), apothem (a), or radius (r) in the respective input field. Ensure the value is positive.
- Calculate: Click the "Calculate" button or simply change any input value after the initial entry.
- Read the Results: The calculator will instantly display:
- The primary result: The Regular Polygon Area.
- Intermediate values: Perimeter, Interior Angle, Exterior Angle, and the calculated values for the side, apothem, and radius based on your input.
- The formula used for the calculation.
- Reset (Optional): Click "Reset" to clear the inputs and results to their default values.
- Copy Results (Optional): Click "Copy Results" to copy the main area, intermediate values, and input parameters to your clipboard.
The results help you understand the space covered by your regular polygon. The intermediate values provide more geometric insights.
Key Factors That Affect Regular Polygon Area Results
- Number of Sides (n): For a fixed side length, apothem, or radius, increasing the number of sides generally increases the area, approaching the area of a circle with the same radius/apothem as n becomes very large.
- Side Length (s): If n is fixed, the area increases with the square of the side length. Doubling the side length quadruples the Regular Polygon Area.
- Apothem (a): If n is fixed, the area increases with the square of the apothem.
- Radius (r): If n is fixed, the area increases with the square of the radius.
- Units Used: Ensure consistency in units. If you input the side length in cm, the area will be in cm².
- Angle Measurement (Radians/Degrees): The formulas use trigonometric functions (tan, sin) where the angle (π/n or 2π/n) is typically in radians within the calculation, though we often think of interior/exterior angles in degrees. Our calculator handles this internally.
Understanding these factors helps in predicting how the Regular Polygon Area changes with different parameters. For more on perimeters, see our perimeter of polygon tool.
Frequently Asked Questions (FAQ)
- What is a regular polygon?
- A regular polygon is a polygon that is both equilateral (all sides are equal in length) and equiangular (all interior angles are equal in measure).
- What is the difference between apothem and radius?
- The apothem is the distance from the center of a regular polygon to the midpoint of one of its sides. The radius (or circumradius) is the distance from the center to one of its vertices (corners).
- Can I calculate the area of an irregular polygon with this calculator?
- No, this calculator is specifically for regular polygons. Irregular polygons require different methods, such as dividing them into triangles or using the Shoelace formula if coordinates are known.
- What is the minimum number of sides a polygon can have?
- A polygon must have at least 3 sides (a triangle).
- How does the area change as the number of sides increases, keeping the radius constant?
- As the number of sides (n) of a regular polygon inscribed in a circle of constant radius increases, its area gets closer and closer to the area of the circle (πr²). Our circle area calculator can help here.
- What if I only know the perimeter and the number of sides?
- If you know the perimeter (P) and number of sides (n), you can find the side length (s = P/n) and then use the formula for Regular Polygon Area based on n and s.
- Why does the calculator ask for only one dimension (side, apothem, or radius) besides the number of sides?
- For a regular polygon, knowing the number of sides and any one of these three dimensions (side length, apothem, or radius) is sufficient to determine all other properties, including the area, due to the geometric relationships between them.
- Are the angles in the formulas in degrees or radians?
- The trigonometric functions (tan, sin) in the formulas expect the angle (like π/n) to be in radians. The calculator handles this conversion internally when showing interior/exterior angles in degrees.
For basic shapes, you might find our triangle area calculator or square area calculator useful.
Related Tools and Internal Resources
- Area Calculators: A collection of calculators for various geometric shapes.
- Perimeter Calculator: Calculate the perimeter of different shapes, including regular polygons.
- Triangle Area Calculator: Find the area of a triangle using different formulas.
- Square Area Calculator: Quickly find the area and perimeter of a square.
- Circle Area Calculator: Calculate the area and circumference of a circle.
- Geometry Formulas: A reference page for common geometry formulas.