Finding Lenght Of Sides Pentagon Calculator

Calculate the side length of a regular pentagon based on its apothem, radius (circumradius), or area. Select the known value and enter it below.

Pentagon Properties Chart & Table

Side Length (s)	Apothem (a)	Radius (R)	Area (A)	Perimeter (P)

Table of Regular Pentagon Properties for Different Side Lengths

What is a Pentagon Side Length Calculator?

A Pentagon Side Length Calculator is a tool designed to determine the length of the sides of a regular pentagon when other properties like its apothem (inradius), radius (circumradius), or area are known. A regular pentagon has five equal sides and five equal internal angles (each 108°). This calculator simplifies the geometric calculations involved.

This calculator is useful for students studying geometry, architects, engineers, designers, and anyone needing to work with the dimensions of a regular pentagon. It avoids manual calculations using trigonometric functions and square roots, providing quick and accurate results. A common misconception is that you need to know an angle other than the standard 108° for a regular pentagon; for a regular one, the angles are fixed.

Pentagon Side Length Formula and Mathematical Explanation

The side length (s) of a regular pentagon can be calculated from its apothem (a), radius (R), or area (A) using the following formulas, derived from the geometry of the pentagon and trigonometric relationships:

• From Apothem (a): `s = 2 * a * tan(π/5) = 2 * a * tan(36°) ≈ 2 * a * 0.7265`
• From Radius (R): `s = 2 * R * sin(π/5) = 2 * R * sin(36°) ≈ 2 * R * 0.5878`
• From Area (A): `s = sqrt((4 * A * tan(π/5)) / 5) = sqrt((4 * A * 0.7265) / 5) ≈ sqrt(0.5812 * A)`

Where `π/5` radians is equal to 36 degrees. The constants `tan(36°)` and `sin(36°)` are derived from the geometry of the pentagon, which can be divided into five congruent isosceles triangles from its center.

Variables Table

Variable	Meaning	Unit	Typical Range
s	Side length	Length (e.g., cm, m, inches)	> 0
a	Apothem (inradius)	Length (e.g., cm, m, inches)	> 0
R	Radius (circumradius)	Length (e.g., cm, m, inches)	> 0
A	Area	Area (e.g., cm², m², inches²)	> 0
P	Perimeter	Length (e.g., cm, m, inches)	> 0

Practical Examples (Real-World Use Cases)

Example 1: Given Apothem

Suppose you are designing a small garden bed in the shape of a regular pentagon and know the shortest distance from the center to any side (apothem) is 3 meters.

• Input: Apothem (a) = 3 m
• Calculation: `s = 2 * 3 * tan(36°) ≈ 6 * 0.7265 ≈ 4.359 m`
• Result: The side length of the pentagon garden bed would be approximately 4.36 meters.

Example 2: Given Area

An architect is working with a pentagonal window that needs to have an area of 2 square meters.

• Input: Area (A) = 2 m²
• Calculation: `s = sqrt((4 * 2 * tan(36°)) / 5) ≈ sqrt((8 * 0.7265) / 5) ≈ sqrt(1.1624) ≈ 1.078 m`
• Result: Each side of the pentagonal window would be approximately 1.08 meters long. Using our Pentagon Side Length Calculator makes this quick.

How to Use This Pentagon Side Length Calculator

1. Select Input Type: Choose whether you know the Apothem, Radius, or Area by selecting the corresponding radio button.
2. Enter Known Value: Input the value of the apothem, radius, or area into the active input field. Ensure the value is positive.
3. Calculate: The calculator will automatically update the results as you type, or you can click the "Calculate" button.
4. View Results: The calculated Side Length (s) will be displayed prominently, along with the Perimeter (P), and the other two properties (Apothem, Radius, Area) that were not used as input. The internal angle is always 108° for a regular pentagon.
5. Reset: Click "Reset" to clear the inputs and results to default values.
6. Copy Results: Click "Copy Results" to copy the main results and input to your clipboard.

Understanding the results helps in various applications, from design and construction to educational purposes. The Pentagon Side Length Calculator provides all key dimensions based on one initial measurement.

Key Factors That Affect Pentagon Side Length Results

• Input Value: The numerical value of the apothem, radius, or area directly determines the side length. Larger inputs result in larger side lengths.
• Chosen Property: Whether you start with apothem, radius, or area changes the formula used, but for a given regular pentagon, they are all interrelated.
• Accuracy of tan(36°) and sin(36°): The calculator uses precise values for these trigonometric constants. Using rounded values in manual calculations can lead to slight differences.
• Units: Ensure consistency in units. If you input apothem in cm, the side length will be in cm.
• Regularity: This calculator assumes a regular pentagon (all sides and angles equal). It won't work for irregular pentagons.
• Measurement Precision: The accuracy of your input measurement (apothem, radius, or area) will directly impact the precision of the calculated side length. Our geometry formulas guide can help.

Frequently Asked Questions (FAQ)

Q: What is a regular pentagon? A: A regular pentagon is a five-sided polygon with all sides of equal length and all internal angles equal (108°).

Q: Can I use this calculator for an irregular pentagon? A: No, this Pentagon Side Length Calculator is specifically for regular pentagons where all sides are equal.

Q: What is the difference between apothem and radius? A: The apothem is the distance from the center to the midpoint of a side, while the radius (or circumradius) is the distance from the center to a vertex. See our apothem calculator.

Q: How are the formulas derived? A: The formulas come from dividing the pentagon into five congruent isosceles triangles from the center, then using trigonometry within one of these triangles (or half of one, forming a right-angled triangle).

Q: What units can I use? A: You can use any unit of length (cm, m, inches, feet, etc.) for apothem and radius, and the corresponding square units for area (cm², m², etc.). The side length will be in the same unit of length as the input.

Q: Why is the internal angle always 108°? A: For any regular n-gon, the internal angle is (n-2)*180/n degrees. For a pentagon (n=5), this is (5-2)*180/5 = 3*180/5 = 540/5 = 108°.

Q: What if I enter zero or a negative number? A: The calculator will show an error or produce non-sensical results, as dimensions like apothem, radius, area, and side length must be positive.

Q: How accurate is this Pentagon Side Length Calculator? A: The calculator uses standard mathematical formulas and high-precision values for constants, providing very accurate results based on your input.