Calculated Results
Results appear here immediately after calculation.
Scale Visualization
The chart compares the main geometric outputs for the current regular pentagon.
Calculator Inputs
Choose one known property, enter its value, and the calculator derives the full regular pentagon set.
Example Data Table
These sample entries show how one known value determines every other regular pentagon dimension.
| Known Type | Input | Side | Perimeter | Area | Diagonal |
|---|---|---|---|---|---|
| Side length | 10.000 cm | 10.000 cm | 50.000 cm | 172.048 cm² | 16.180 cm |
| Apothem | 6.882 cm | 10.000 cm | 50.000 cm | 172.048 cm² | 16.180 cm |
| Circumradius | 8.507 cm | 10.000 cm | 50.000 cm | 172.048 cm² | 16.180 cm |
Formula Used
Perimeter:
P = 5sApothem:
a = s / (2 tan(π/5))Circumradius:
R = s / (2 sin(π/5))Diagonal:
d = φs, where φ = (1 + √5) / 2Area:
A = (5s²) / (4 tan(π/5)) = 0.5Pa
From perimeter:
s = P / 5From area:
s = √((4A tan(π/5)) / 5)From apothem:
s = 2a tan(π/5)From circumradius:
s = 2R sin(π/5)From diagonal:
s = d / φ
The calculator uses radian-based trigonometric functions for consistency and precise geometry outputs.
How to Use This Calculator
- Select the property you already know.
- Enter a positive numeric value for that property.
- Add an optional unit label such as cm or m.
- Choose the decimal precision you want displayed.
- Apply an optional scale factor to resize the pentagon instantly.
- Press Calculate Pentagon to view results above the form.
- Review the plotted comparison chart and exported summary options.
Frequently Asked Questions
1. What does this regular pentagon calculator solve?
It solves side length, perimeter, area, apothem, circumradius, diagonal, and standard angle measures for a regular pentagon using one known input.
2. Why does one measurement determine everything else?
A regular pentagon has equal sides and equal angles. That fixed geometry means every major dimension scales from a single base measurement.
3. What is the difference between apothem and circumradius?
The apothem runs from the center to the midpoint of a side. The circumradius runs from the center to a vertex.
4. How is the diagonal related to the side?
In a regular pentagon, the diagonal equals the side multiplied by the golden ratio. This creates the famous internal pentagram relationship.
5. Can I use area as the starting input?
Yes. The calculator reverses the standard area equation to recover the side length, then computes all remaining dimensions automatically.
6. What angles are constant in a regular pentagon?
Each interior angle is 108 degrees, each exterior angle is 72 degrees, and each central angle is also 72 degrees.
7. What does the scale factor do?
It multiplies the solved side length before final reporting. This helps compare a base pentagon with a resized version quickly.
8. Are the exports suitable for reports or homework?
Yes. The CSV file is useful for spreadsheets, while the PDF export captures the visible result summary for sharing or printing.