15 Gon Area Calculator

A 15-gon is a 15-sided polygon. Calculating its area requires knowing the side length and the apothem (the distance from the center to the midpoint of any side). This calculator provides precise area calculations for regular 15-gons.

What is a 15-gon?

A 15-gon is a polygon with fifteen straight sides and fifteen vertices. When all sides and angles are equal, it's called a regular 15-gon. Regular 15-gons are common in geometry, architecture, and engineering.

Key properties of regular 15-gons:

Each interior angle is 156°
Each exterior angle is 24°
Sum of interior angles is 2160°
Sum of exterior angles is 360°

How to calculate 15-gon area

To calculate the area of a regular 15-gon, you need two key measurements:

The length of one side (s)
The apothem (a) - the distance from the center to the midpoint of any side

The formula for the area (A) of a regular polygon is:

A = (1/2) × perimeter × apothem

For a 15-gon: A = (1/2) × (15 × s) × a

Alternatively, you can use the side length and the interior angle to calculate the apothem first.

The formula

The area of a regular 15-gon can be calculated using these formulas:

Area = (1/2) × perimeter × apothem

Perimeter = 15 × side length

Apothem = side length / (2 × tan(π/15))

Where:

π (pi) is approximately 3.14159
tan is the tangent function
π/15 is the central angle in radians

Note: These formulas assume a regular 15-gon where all sides and angles are equal. Irregular 15-gons require more complex calculations.

Worked example

Let's calculate the area of a regular 15-gon with each side measuring 5 units.

Calculate the perimeter: 15 × 5 = 75 units
Calculate the apothem: 5 / (2 × tan(π/15)) ≈ 5 / (2 × 0.3648) ≈ 6.84 units
Calculate the area: (1/2) × 75 × 6.84 ≈ 256.5 square units

Using our calculator with these inputs would give you approximately 256.5 square units.

FAQ

What's the difference between a 15-gon and a pentadecagon?

Both terms refer to the same shape - a polygon with fifteen sides. "Pentadecagon" is the more formal geometric term, while "15-gon" is more commonly used in everyday language.

Can I calculate the area of an irregular 15-gon?

Yes, but it requires knowing the coordinates of all vertices or using more advanced geometric techniques. Our calculator is designed for regular 15-gons only.

What's the largest regular 15-gon that can fit in a circle?

The largest regular 15-gon that can fit inside a circle is called a circumscribed 15-gon. Its side length is determined by the circle's radius and the properties of regular polygons.