Last updated: August 5, 2026
Pentagon Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Quick Answer
A regular pentagon can be solved from one side length using A = 5s² ÷ (4 tan(π ÷ 5)), P = 5s, and d = φ × s. This calculator returns the area, perimeter, diagonal, apothem, circumradius, and fixed 108-degree interior angle for a regular pentagon.
For a regular pentagon, the perimeter equals five times the side length, the diagonal equals the golden ratio times the side, and the area comes from five times side squared divided by four times tangent of pi over five.
Key Takeaways
- A regular pentagon is fully determined by one side length.
- The diagonal-to-side ratio equals the golden ratio φ.
- Area can be derived from regular-polygon trigonometry or one half times perimeter times apothem.
- Apothem and circumradius are different center-based measurements.
- Linear dimensions scale with side length, but area scales with side length squared.
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
A = 5s² ÷ (4 tan(π ÷ 5)), P = 5s, d = φ × s
Where:
- s=Side length(units)
- A=Area(square units)
- P=Perimeter(units)
- d=Diagonal(units)
- φ=Golden ratio
Worked Examples
Regular pentagon with side 5
A clean classroom example using an integer side length.
- 1Perimeter equals 5 × 5 = 25.
- 2Diagonal equals φ × 5 ≈ 8.09017.
- 3The tangent-based area formula gives 43.011935 square units.
- 4Apothem and circumradius stay in the same linear unit as the side.
Sign panel with side 12.5
A fabrication-style problem where one edge dimension drives the full layout.
- 1Perimeter equals 62.5.
- 2Diagonal equals 20.225425.
- 3Area equals 268.824594 square units.
- 4Apothem and circumradius support center-based layout checks.
Scaled sketch with side 3.2
A smaller regular pentagon still follows the same fixed geometry.
- 1Perimeter equals 16.
- 2Diagonal equals 5.177709.
- 3Area equals 17.617688 square units.
- 4Interior angle remains 108 degrees because regularity fixes it.
Introduction
The Pentagon Calculator is built for the special case of a regular pentagon, where all five sides and all five angles are equal. That symmetry means one side length is enough to recover the perimeter, area, diagonal, apothem, circumradius, and the fixed interior angle of 108 degrees. The calculator is useful for geometry homework, drafting, fabrication, and design because it connects those measurements in one consistent unit system and makes the golden-ratio diagonal relationship easy to verify.
What defines a regular pentagon
A regular pentagon has five equal sides and five equal interior angles. Because of that symmetry, every major measurement can be traced back to one side length, which is why this calculator only needs a single geometric input.
Five equal sides create rotational symmetry.
Each interior angle is 108 degrees.
The central geometry can be split into five congruent triangles.
One side length determines every other regular-pentagon dimension.
How the formulas connect
The perimeter is a direct multiple of the side length, while the apothem and circumradius come from the right triangles formed inside the pentagon. Area can then be found through the regular-polygon identity one half times perimeter times apothem, which is equivalent to the tangent-based formula shown at the top.
Perimeter scales linearly with the side.
Apothem and circumradius come from center-based triangles.
Area uses the same side length in a trigonometric relationship.
The formulas reinforce each other rather than standing alone.
Why the diagonal uses the golden ratio
One of the most famous facts about a regular pentagon is that its diagonal divided by its side equals the golden ratio. That makes the diagonal a powerful quick-check output: if a supposed regular pentagon has a very different side-to-diagonal proportion, either the drawing is irregular or the measurement is off.
Diagonal ≈ 1.618 × side.
The same ratio appears in pentagrams.
Golden-ratio structure is a hallmark of regular pentagon geometry.
The diagonal output is a strong drafting sanity check.
How to use the calculator
Measure one side of the regular pentagon, enter it in a consistent unit, and read the outputs according to your task. Use perimeter for edge length, area for coverage, diagonal for internal spacing, apothem for area derivations, and circumradius for center-based construction work.
Enter a positive side length only.
Keep all linear units consistent.
Interpret area separately because it uses squared units.
Use center-based outputs for geometric constructions.
Apothem versus circumradius
The apothem is the segment from the center to the midpoint of a side, drawn at a right angle. The circumradius is the segment from the center to a vertex. Both are radii in a loose sense, but they support different tasks and should not be interchanged in geometry work.
Apothem reaches a side midpoint.
Circumradius reaches a vertex.
Apothem appears directly in the area identity.
Circumradius matters in inscribed-circle layouts.
Common mistakes in pentagon calculations
The most common mistake is applying regular-polygon formulas to an irregular five-sided figure. People also confuse the diagonal with the side when a star shape is drawn inside the pentagon, and they often mix up linear units with square units when discussing area.
Regular formulas do not apply to irregular pentagons.
Area units must be squared.
The diagonal is not the same as the side.
Apothem and circumradius solve different problems.
Where pentagon measurements are used
Regular pentagons appear in ornamental layouts, logo drafts, pattern studies, classroom constructions, and some fabrication tasks. A single side measurement often has to be translated into coverage, radius, or spacing values, which is why a connected calculator is more useful than isolated formulas.
Logo and pattern drafting.
Material estimates from area or perimeter.
Geometry instruction involving symmetry and the golden ratio.
Layout verification from a single edge measurement.
Reference values for common side lengths
A few benchmark side lengths make it easier to estimate what a full solution should look like before you measure carefully. Notice how all linear values double when the side doubles, while the area grows with the square of the side length.
Linear outputs scale directly with the side length.
Area grows quadratically.
Diagonal checks regularity through the golden ratio.
Reference values are useful for sketches and quick estimates.
| Side s | Perimeter P | Diagonal d | Area A |
|---|---|---|---|
| 2 | 10 | 3.236068 | 6.88191 |
| 5 | 25 | 8.09017 | 43.011935 |
| 8 | 40 | 12.944272 | 110.110548 |
| 10 | 50 | 16.18034 | 172.04774 |
Quick Reference Card
Regular Pentagon Quick Reference
Quick reference • Pentagon Calculator
Perimeter = 5s; area = 5s² ÷ (4 tan(π ÷ 5)); diagonal = φ × sValid range: Use any positive side length for a regular pentagon.
Common Values
⚠ Watch Out
- •These formulas apply only to regular pentagons.
- •Keep side, diagonal, and radius values in the same linear unit.
- •Area is reported in square units, not plain units.
- •Do not confuse the apothem with the circumradius.
Pro Tips
- →Use the diagonal output to check whether a drawing is truly regular.
- →Estimate diagonal quickly as about 1.618 times the side.
- →Use the apothem when checking area by hand.
- →Center-based constructions usually need the circumradius more than the perimeter.
FAQs
What does this calculator assume?
It assumes the polygon is a **regular pentagon**. If the side lengths or angles are not equal, the formulas used here no longer describe the full shape correctly.
Why is the diagonal longer than the side?
A diagonal skips one vertex, so it spans farther across the pentagon than a side does. In a regular pentagon it is always the golden ratio times the side.
What is the difference between apothem and circumradius?
The apothem reaches a side midpoint and supports the area calculation, while the circumradius reaches a vertex and supports inscribed-circle layout work.
Can I use inches, centimeters, or meters?
Yes. Any consistent unit works for the side length. Linear outputs stay in that unit, while area is returned in that unit squared.
Why is the interior angle always 108 degrees?
The interior angles of any pentagon add to 540 degrees. A regular pentagon has five equal interior angles, so each one is 540 ÷ 5 = 108 degrees.
How is the area formula derived?
You can split the regular pentagon into five congruent triangles from the center and then express each triangle with tangent-based relationships. That leads to the compact formula shown in the calculator.
When is this calculator most useful?
It is most useful when one clean side length is known and you need the rest of the regular-pentagon geometry for design, drafting, or teaching.