Easily calculate the area of a regular polygon using side length and number of sides on Examples.com. Ideal for geometry, design, and construction applications that require accurate polygonal area measurements.

**Formula: **Area = (n × side^{2}) / (4 × tan(π / n))

The Area of a Regular Polygon refers to the total space enclosed by a polygon with equal sides and angles. Regular polygons can have any number of sides, such as triangles, squares, pentagons, or hexagons. The area is calculated using a specific formula based on the side length and number of sides. This calculation is important in fields like geometry, architecture, and design, where precise area measurements are needed for materials, tiling, and spatial planning. Regular polygons are often used in construction and tiling due to their symmetrical shape, making them efficient for covering spaces without gaps or overlaps.

## How to Find the Area of a Regular Polygon

### Step 1: Identify the Side Length

Determine the **side length** of the regular polygon. All sides of a regular polygon are equal in length.

### Step 2: Count the Number of Sides

Count the total **number of sides** (n) of the polygon. A regular polygon can have any number of sides greater than or equal to 3 (e.g., triangle, square, pentagon, etc.).

### Step 3: Use the Formula

Apply the formula provided: Area=n×side^{2}/4×tan(π/n)

Where **n** is the number of sides, and **side** is the length of each side.

### Step 4: Input Values

Input the **side length** and **number of sides** into their respective fields on the calculator.

### Step 5: Calculate the Area

Click the **Calculate** button, and the area of the regular polygon will be displayed based on the values you provided.

## Area Of Regular Polygon Formula

The formula for the **Area of a Regular Polygon** is: Area=n×side^{2}/4×tan(π/n)

Where:

**n**is the number of sides of the polygon.**side**is the length of one side of the polygon.

## Properties of Area of a Regular Polygon

**Equal Sides and Angles**:

In a regular polygon, all sides and interior angles are equal, making the area easier to calculate due to the symmetry of the shape.**Depends on Number of Sides (n)**:

The area of a regular polygon is influenced by the number of sides. As the number of sides increases, the polygon approaches the shape of a circle, resulting in a larger area.**Proportional to Side Length**:

The area is proportional to the square of the side length, meaning that if the side length doubles, the area increases fourfold.**Involves Angle Relations**:

The calculation of the area involves trigonometric functions, which account for the angles between the sides of the polygon and help determine the relationship between the side length and the overall shape.**Measured in Square Units**:

The area of a regular polygon is expressed in square units, such as square meters, square centimeters, or square feet, depending on the units used for the side length.**Approaches a Circle as Sides Increase**:

As the number of sides increases, a regular polygon becomes more circular in shape. A polygon with an infinite number of sides would be indistinguishable from a circle.**Apothem Use in Area Calculation**:

In some cases, the apothem, or the perpendicular distance from the center to a side, can also be used to calculate the area of a regular polygon.**Perimeter and Apothem Method**:

Another method to find the area is by using the perimeter and apothem, which is useful in certain geometric problems or real-world applications.**Only for Regular Polygons**:

This method of area calculation is specific to regular polygons where all sides and angles are equal. Irregular polygons require different approaches.**Applications in Design and Engineering**:

The area of regular polygons is important in fields like architecture and engineering, where precise geometric patterns are used in tiling, construction, and design.

## Area of a Regular Polygon Examples

### Example 1:

**Given:**

Side length = 4 m

Number of sides = 6 (Hexagon)

**Solution:** Area=6×4^{2}/4×tan(π/6) = 41.57 m^{2}

### Example 2:

**Given:**

Side length = 5 cm

Number of sides = 8 (Octagon)

**Solution:** Area=8×5^{2}/4×tan(π/8) = 120.71 cm^{2}

### Example 3:

**Given:**

Side length = 3 m

Number of sides = 5 (Pentagon)

**Solution:** Area=5×3^{2}/4×tan(π/5) = 15.48 m^{2}

### Example 4:

**Given:**

Side length = 6 ft

Number of sides = 7 (Heptagon)

**Solution:** Area=7×6^{2}/4×tan(π/7) = 130.82 ft^{2}

### Example 5:

**Given:**

Side length = 2 m

Number of sides = 10 (Decagon)

**Solution:** Area=10×2^{2}/4×tan(π/10) = 30.77 m^{2}

## Does the number of sides affect the area of a regular polygon?

Yes, the number of sides directly affects the area. As the number of sides increases, the polygon becomes more circular, and the area becomes larger.

## What happens to the area if the side length is doubled?

If the side length of a regular polygon is doubled, the area increases by a factor of four. This is because the area is proportional to the square of the side length.

## Is there a limit to the number of sides a regular polygon can have?

There is no theoretical limit to the number of sides a regular polygon can have. As the number of sides increases, the polygon becomes increasingly similar to a circle.

## Does the area of a regular polygon depend on the interior angles?

Yes, the interior angles affect the area calculation because they influence the shape of the polygon. This is accounted for by the tangent function in the area formula.

## Is the area of a regular polygon always larger than an irregular polygon with the same perimeter?

Not necessarily. The shape and angles of the polygon affect the area. A regular polygon is efficient in maximizing the enclosed area, but an irregular polygon can have a larger or smaller area depending on its shape.

## What is the easiest way to calculate the area of a polygon with many sides?

For polygons with many sides (e.g., 20 or more), the regular polygon formula becomes less practical for manual calculations. Instead, using software tools or a calculator with the appropriate formula is easier.