Data Source & Methodology

This calculator computes the properties of a regular nonagon, which is a nine-sided polygon with all sides of equal length and all internal angles equal. The calculations are derived from standard trigonometric principles by dividing the nonagon into 9 identical isosceles triangles.

• Authoritative Source: The formulas are based on the **fundamental properties of regular polygons** and **trigonometric laws** (SOH CAH TOA). The specific angles are derived from the central angle of the nonagon ($360^\circ / 9 = 40^\circ$).
• Methodology: "Tutti i calcoli si basano rigorosamente sulle formule e sui dati forniti da questa fonte." The core method involves solving the small right triangle formed by the **apothem ($r$)**, the **circumradius ($R$)**, and **half the side length ($a/2$)**. The central angle of this small triangle is $\theta = (360^\circ / 9) / 2 = 20^\circ$ (or $\pi/9$ radians).

The Formulas Explained

All properties of a regular nonagon can be found if you know just one of its values. The primary variable is the **side length ($a$)**. The formulas use the internal angle $\theta = 20^\circ$ or $(\pi/9 \text{ rad})$.

1. Calculations from Side ($a$)

These are the base formulas used to find all other properties.

Apothem ($r$):

Circumradius ($R$):

Area ($A$):

Perimeter ($P$):

2. Reverse Formulas (Solving for $a$)

This calculator uses the following formulas to find the side ($a$) if you provide a different property.

From Area ($A$):

From Apothem ($r$):

From Circumradius ($R$):

Glossary of Variables

• Side ($a$): The length of one of the 9 equal sides of the nonagon.
• Apothem ($r$): The inradius. The radius of the inscribed circle; the perpendicular distance from the center to any side.
• Circumradius ($R$): The radius of the circumscribed circle; the distance from the center to any vertex.
• Area ($A$): The total space enclosed by the nonagon's sides.
• Perimeter ($P$): The total distance around the nonagon's boundary ($9 \times a$).
• Internal Angle: The angle at each vertex, which is always $140^\circ$ for a regular nonagon.

How It Works: A Step-by-Step Example

Let's find the properties of a nonagon that has an **Apothem (r) of 10 inches**.

1. Select the "Given" Value:

   In the calculator, choose "Apothem (r)" from the dropdown and enter "10". Select "in" as the unit.

2. Step 1: Find Side ($a$) from Apothem ($r$):

   The calculator first finds the side length using the reverse formula. We use $\tan(20^\circ) \approx 0.36397$.

   $$ a = 2r \tan(20^\circ) = 2 \times 10 \times 0.36397 = 7.2794 \text{ in} $$
3. Step 2: Calculate Other Properties from Side ($a$):

   Now that $a = 7.2794 \text{ in}$, the calculator finds all other values.

   Perimeter ($P$):

   $$ P = 9a = 9 \times 7.2794 = 65.5146 \text{ in} $$

   Area ($A$):

   $$ A = \frac{9 a^2}{4 \tan(20^\circ)} = \frac{9 \times (7.2794)^2}{4 \times 0.36397} \approx 327.57 \text{ in}^2 $$

   Circumradius ($R$):

   $$ R = \frac{a}{2 \sin(20^\circ)} = \frac{7.2794}{2 \times 0.34202} \approx 10.6418 \text{ in} $$

Result: A nonagon with a 10-inch apothem has a side of 7.28 in, an area of 327.57 in², a perimeter of 65.51 in, and a circumradius of 10.64 in.

Frequently Asked Questions (FAQ)

Data Source & Methodology

This calculator computes the properties of a regular nonagon, which is a nine-sided polygon with all sides of equal length and all internal angles equal. The calculations are derived from standard trigonometric principles by dividing the nonagon into 9 identical isosceles triangles.

• Authoritative Source: The formulas are based on the **fundamental properties of regular polygons** and **trigonometric laws** (SOH CAH TOA). The specific angles are derived from the central angle of the nonagon ($360^\circ / 9 = 40^\circ$).
• Methodology: "Tutti i calcoli si basano rigorosamente sulle formule e sui dati forniti da questa fonte." The core method involves solving the small right triangle formed by the **apothem ($r$)**, the **circumradius ($R$)**, and **half the side length ($a/2$)**. The central angle of this small triangle is $\theta = (360^\circ / 9) / 2 = 20^\circ$ (or $\pi/9$ radians).

The Formulas Explained

All properties of a regular nonagon can be found if you know just one of its values. The primary variable is the **side length ($a$)**. The formulas use the internal angle $\theta = 20^\circ$ or $(\pi/9 \text{ rad})$.

1. Calculations from Side ($a$)

These are the base formulas used to find all other properties.

Apothem ($r$):

Circumradius ($R$):

Area ($A$):

Perimeter ($P$):

2. Reverse Formulas (Solving for $a$)

This calculator uses the following formulas to find the side ($a$) if you provide a different property.

From Area ($A$):

From Apothem ($r$):

From Circumradius ($R$):

Glossary of Variables

• Side ($a$): The length of one of the 9 equal sides of the nonagon.
• Apothem ($r$): The inradius. The radius of the inscribed circle; the perpendicular distance from the center to any side.
• Circumradius ($R$): The radius of the circumscribed circle; the distance from the center to any vertex.
• Area ($A$): The total space enclosed by the nonagon's sides.
• Perimeter ($P$): The total distance around the nonagon's boundary ($9 \times a$).
• Internal Angle: The angle at each vertex, which is always $140^\circ$ for a regular nonagon.

How It Works: A Step-by-Step Example

Let's find the properties of a nonagon that has an **Apothem (r) of 10 inches**.

1. Select the "Given" Value:

   In the calculator, choose "Apothem (r)" from the dropdown and enter "10". Select "in" as the unit.

2. Step 1: Find Side ($a$) from Apothem ($r$):

   The calculator first finds the side length using the reverse formula. We use $\tan(20^\circ) \approx 0.36397$.

   $$ a = 2r \tan(20^\circ) = 2 \times 10 \times 0.36397 = 7.2794 \text{ in} $$
3. Step 2: Calculate Other Properties from Side ($a$):

   Now that $a = 7.2794 \text{ in}$, the calculator finds all other values.

   Perimeter ($P$):

   $$ P = 9a = 9 \times 7.2794 = 65.5146 \text{ in} $$

   Area ($A$):

   $$ A = \frac{9 a^2}{4 \tan(20^\circ)} = \frac{9 \times (7.2794)^2}{4 \times 0.36397} \approx 327.57 \text{ in}^2 $$

   Circumradius ($R$):

   $$ R = \frac{a}{2 \sin(20^\circ)} = \frac{7.2794}{2 \times 0.34202} \approx 10.6418 \text{ in} $$

Result: A nonagon with a 10-inch apothem has a side of 7.28 in, an area of 327.57 in², a perimeter of 65.51 in, and a circumradius of 10.64 in.

Frequently Asked Questions (FAQ)