Pentagon Calculator
Calculate the area, perimeter, and apothem of a regular pentagon from its side length.
Minimum: 0
Length of one side of the regular pentagon
Results
Area
43.0119 sq units
Perimeter
25 units
Apothem
3.441 units
How to Use This Calculator
Enter your input values
Fill in all required input fields for the Pentagon Calculator. Follow the units and input format printed next to each field. Where a unit selector is available, choose the units that match your data.
Review your inputs
Double-check that all values are correct and match the units or format shown for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.
Read the results
The Pentagon Calculator computes the output in your browser and displays the results, with units clearly labeled, as soon as they are ready — no data sent to a server.
Explore parameter sensitivity
Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.
When to Use This Calculator
- •Use the Pentagon Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
- •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
- •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
- •Use it to explore the behavior of mathematical functions across a range of inputs.
Related Calculators
Octagon Calculator
Calculate the area, perimeter, and apothem of a regular octagon from its side length.
Hexagon Calculator
Calculate the area, perimeter, and other properties of a regular hexagon from its side length.
Equilateral Triangle Calculator
Calculate the area, perimeter, and height of an equilateral triangle from its side length.
Regular Polygon Area Calculator
Calculate the area of any regular polygon given the number of sides and side length using A = n*s^2/(4*tan(PI/n)). Works for equilateral triangles, squares, pentagons, hexagons, and any regular n-gon used in architecture, tiling, and engineering.
Isosceles Triangle Calculator
Calculate the area, perimeter, and height of an isosceles triangle from its equal side and base.
Area of a Rectangle Calculator
Calculate the area of a rectangle from its length and width. Supports any unit of measurement with instant results.
About Pentagon Calculator
A regular pentagon is a five-sided polygon with all sides equal and all interior angles equal to 108 degrees. The pentagon is a shape of deep mathematical significance, intimately connected to the golden ratio φ = (1 + √5)/2 ≈ 1.618. The ratio of a diagonal to a side in a regular pentagon equals the golden ratio, a relationship that fascinated ancient Greek mathematicians. Pentagons appear in architecture (the Pentagon building in Washington, D.C.), nature (starfish, some flowers), and art. This calculator computes the area, perimeter, and apothem of a regular pentagon given its side length. The pentagon cannot tile the plane alone, unlike triangles, squares, and hexagons, which led to centuries of research into pentagonal tilings and eventually contributed to the discovery of quasicrystals.
The Math Behind It
Formula Reference
Regular Pentagon Area
A = (1/4) × √(5(5 + 2√5)) × s²
Variables: s = side length
Pentagon Apothem
a = s / (2 × tan(π/5))
Variables: s = side length
Worked Examples
Example 1: Pentagonal table
A table top is a regular pentagon with 0.8 m sides. Find the area.
A ≈ 1.1013 m²
Common Mistakes & Tips
- !Using the wrong interior angle — a regular pentagon has 108° angles, not 100° or 120°.
- !Confusing the apothem with the radius (circumradius) of the pentagon.
- !Applying formulas for regular pentagons to irregular five-sided polygons.
Related Concepts
Used in These Calculators
Calculators that build on or apply the concepts from this page:
Frequently Asked Questions
What is the golden ratio's connection to pentagons?
In a regular pentagon, the ratio of a diagonal to a side equals the golden ratio φ ≈ 1.618. This ratio appears repeatedly in the geometry of pentagons and pentagrams.
Can pentagons tile a flat surface?
Regular pentagons cannot tile the plane alone. However, certain irregular convex pentagons can, and 15 types have been discovered as of 2015.
Add the live Pentagon Calculator to your site. Free to embed, with a link back to MegaCalc.