Pentagon Side Length Calculator
Calculate the side length of a regular pentagon from its apothem, radius (circumradius), or area using our pentagon side length calculator.
Calculate Side Length
What is a Pentagon Side Length Calculator?
A pentagon side length calculator is a tool used to determine the length of one of the five equal sides of a regular pentagon when other properties like the apothem (inradius), radius (circumradius), or area are known. A regular pentagon has five equal sides and five equal interior angles (each 108°).
This calculator is useful for students, engineers, designers, and anyone working with geometric shapes, particularly regular pentagons. It eliminates the need for manual calculations using trigonometric formulas. Common misconceptions include assuming all pentagons are regular or that side length can be found from just one angle (which is fixed for regular pentagons).
Pentagon Side Length Calculator 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 trigonometry:
- From Apothem (a): The apothem is the distance from the center to the midpoint of a side. The formula is: `s = 2 * a * tan(π/5)` where `π/5` radians is 36 degrees.
- From Radius (r): The radius (or circumradius) is the distance from the center to a vertex. The formula is: `s = 2 * r * sin(π/5)`
- From Area (A): The area of a regular pentagon is given by `A = (5 * s^2) / (4 * tan(π/5))`. Rearranging for s: `s = sqrt((4 * A * tan(π/5)) / 5)`
The value `tan(π/5) ≈ 0.726542528` and `sin(π/5) ≈ 0.587785252` are constants derived from the 36° angle central to the pentagon's geometry.
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 |
| π/5 | Angle (36°) | Radians (or degrees) | Constant |
Practical Examples (Real-World Use Cases)
Let's see how the pentagon side length calculator works with practical examples.
Example 1: Given Apothem
Suppose you have a regular pentagon with an apothem of 10 cm. Using the formula `s = 2 * a * tan(π/5)`:
`s = 2 * 10 * tan(36°) ≈ 2 * 10 * 0.7265 ≈ 14.53 cm`
So, the side length is approximately 14.53 cm.
Example 2: Given Radius
If the radius (circumradius) of a regular pentagon is 12 inches. Using `s = 2 * r * sin(π/5)`:
`s = 2 * 12 * sin(36°) ≈ 2 * 12 * 0.5878 ≈ 14.11 inches`
The side length is approximately 14.11 inches.
Example 3: Given Area
Imagine a regular pentagon with an area of 100 m². Using `s = sqrt((4 * A * tan(π/5)) / 5)`:
`s = sqrt((4 * 100 * 0.7265) / 5) = sqrt(290.6 / 5) = sqrt(58.12) ≈ 7.62 m`
The side length is approximately 7.62 m.
How to Use This Pentagon Side Length Calculator
- Select Input Type: Choose whether you know the Apothem, Radius, or Area from the "Calculate side length from:" dropdown menu.
- Enter Value: Input the known value (apothem, radius, or area) into the corresponding field. Ensure the value is positive.
- View Results: The calculator will automatically update and display the Side Length, Perimeter, and other details as you type or when you click "Calculate".
- Interpret Results: The "Side Length (s)" is the primary result. The perimeter and internal angle are also provided.
- Reset: Click "Reset" to clear the fields and start over with default values.
- Copy: Click "Copy Results" to copy the main outputs to your clipboard.
Our pentagon side length calculator simplifies these calculations instantly.
Key Factors That Affect Pentagon Side Length Results
- Regularity of the Pentagon: The formulas used assume the pentagon is regular (all sides and angles are equal). If it's irregular, these calculations won't apply directly.
- Accuracy of the Known Value: The precision of the calculated side length depends directly on the accuracy of the input apothem, radius, or area.
- Unit Consistency: Ensure you are consistent with units. If you input apothem in cm, the side length will be in cm.
- Chosen Parameter: Whether you start with apothem, radius, or area will determine which formula is used by the pentagon side length calculator.
- Rounding: The values of `tan(π/5)` and `sin(π/5)` involve irrational numbers, so slight rounding occurs, affecting the final result's precision.
- Measurement Errors: If the input value comes from a physical measurement, any error in that measurement will propagate to the side length calculation.
Frequently Asked Questions (FAQ)
Q1: What is a regular pentagon?
A1: A regular pentagon is a five-sided polygon with all sides of equal length and all interior angles equal (108° each).
Q2: What is the apothem of a pentagon?
A2: The apothem (or inradius) is the distance from the center of a regular pentagon to the midpoint of one of its sides. It is perpendicular to the side.
Q3: What is the radius (circumradius) of a pentagon?
A3: The radius (or circumradius) is the distance from the center of a regular pentagon to one of its vertices.
Q4: Can I use this calculator for an irregular pentagon?
A4: No, this pentagon side length calculator is specifically for regular pentagons. Irregular pentagons don't have a single "side length," apothem, or radius in the same way.
Q5: How do I find the area or apothem if I only know the side length?
A5: You can rearrange the formulas. `Area = (5 * s^2) / (4 * tan(π/5))` and `Apothem = s / (2 * tan(π/5))`. You might need our Area Calculator for different shapes.
Q6: What are the units for the side length?
A6: The units of the side length will be the same as the units you used for the apothem or radius, or the square root of the units used for area (e.g., if area is in cm², side is in cm).
Q7: Why is the internal angle always 108° for a regular pentagon?
A7: The sum of interior angles of any n-sided polygon is (n-2) * 180°. For a pentagon (n=5), the sum is (5-2) * 180° = 540°. In a regular pentagon, all 5 angles are equal, so each is 540°/5 = 108°.
Q8: How accurate is this pentagon side length calculator?
A8: The calculator uses standard mathematical formulas and JavaScript's `Math` functions, providing high accuracy. The precision is usually limited by the input value's precision and browser's floating-point arithmetic.
Related Tools and Internal Resources
- Area Calculator: Calculate the area of various shapes, including pentagons if side length is known.
- Perimeter Calculator: Find the perimeter of different geometric figures.
- Polygon Calculator: A general tool for various properties of regular polygons.
- Triangle Calculator: Calculate sides, angles, and area of triangles.
- Circle Calculator: Find circumference, area, and other properties of circles.
- Geometry Formulas: A reference page for common geometry formulas.