Pentagon Calculator

Calculate all properties of a regular pentagon from a single known value.

Select the property you know, enter its value, and the calculator will find the rest.

Examples

Click on an example to load it into the calculator.

Calculate from Side Length

side

If you know the side length of a pentagon, you can find all other properties.

Value: 10

Calculate from Apothem

apothem

If you know the apothem (the distance from the center to a side), you can find all other properties.

Value: 6.88

Calculate from Area

area

If you know the area of a pentagon, you can find its dimensions.

Value: 172.05

Calculate from Perimeter

perimeter

If you know the perimeter of a pentagon, you can easily find its side length and other properties.

Value: 50

Other Titles
Understanding the Pentagon Calculator: A Comprehensive Guide
Explore the geometry of regular pentagons, from basic properties to the mathematical formulas that govern them. This guide covers everything you need to know.

What is a Regular Pentagon? Core Concepts

  • A regular pentagon is a five-sided polygon with equal sides and equal interior angles.
  • Key properties include side length, apothem, perimeter, and area.
  • Understanding these components is crucial for geometric calculations.
A pentagon is a polygon with five sides. A 'regular' pentagon means all its sides are of equal length, and all its interior angles are equal (108°). This regularity simplifies calculations and makes the pentagon a fascinating shape in geometry.
Key Components of a Regular Pentagon
Side (s): The length of one of the five equal sides.
Apothem (a): The perpendicular distance from the center of the pentagon to the midpoint of a side. It is a key element for area calculations.
Perimeter (P): The total length of all sides combined. For a regular pentagon, it's simply 5 times the side length (P = 5s).
Area (A): The amount of space enclosed within the pentagon's sides.
Circumradius (R): The distance from the center to any vertex.

Basic Pentagon Properties

  • If side (s) = 10, then Perimeter (P) = 5 * 10 = 50.
  • All interior angles of a regular pentagon are 108 degrees.
  • All exterior angles are 72 degrees.

Step-by-Step Guide to Using the Pentagon Calculator

  • Select the known value type (side, apothem, area, or perimeter).
  • Input the known value into the designated field.
  • Instantly get all other properties calculated for you.
Our Pentagon Calculator is designed for ease of use. Follow these simple steps to perform your calculations:
How to Use the Calculator
1. Select 'Calculate From': Use the dropdown menu to choose the property of the pentagon you already know. This could be its 'Side (s)', 'Apothem (a)', 'Area (A)', or 'Perimeter (P)'.
2. Enter the Value: In the 'Value' input field, type in the measurement of the property you selected. For example, if you chose 'Side (s)', enter the length of the side.
3. Click 'Calculate': Press the calculate button. The calculator will instantly process the input and display all the other properties of the pentagon in the results section.
4. Reset for a New Calculation: Click the 'Reset' button to clear all fields and start a new calculation.

Practical Usage Scenarios

  • Known Side: Select 'Side (s)', enter '5', click 'Calculate'.
  • Known Area: Select 'Area (A)', enter '100', click 'Calculate'.

Mathematical Formulas and Derivations

  • Learn the formulas used to calculate the properties of a regular pentagon.
  • Understand the relationship between side, apothem, and area.
  • See the derivations based on trigonometry.
The calculations for a regular pentagon are based on trigonometric principles. A regular pentagon can be divided into five congruent isosceles triangles, with the center of the pentagon as their common vertex.
Core Formulas
Given the side length (s):
Perimeter (P) = 5 × s
Apothem (a) = s / (2 × tan(π/5)) ≈ 0.6882 × s
Area (A) = (5 × s²) / (4 × tan(π/5)) ≈ 1.7205 × s²
Circumradius (R) = s / (2 × sin(π/5)) ≈ 0.8507 × s
Given the apothem (a):
Side (s) = 2 × a × tan(π/5) ≈ 1.453 × a
From the side, all other properties can be calculated.
The central angle of each isosceles triangle is 360°/5 = 72°. The apothem bisects this angle and the base side, creating a right-angled triangle with angles 36°, 54°, and 90°. This is the basis for the trigonometric formulas.

Formula Applications

  • For s=10, Area = 1.7205 * 10² = 172.05
  • For a=5, Side = 1.453 * 5 = 7.265

Real-World Applications of Pentagons

  • Pentagons in architecture and design.
  • Natural occurrences of pentagonal shapes.
  • Symbolism and use in branding.
The pentagon shape, while less common than squares or circles, appears in various man-made and natural contexts.
Architecture
The most famous example is the Pentagon building in Arlington, Virginia, the headquarters of the U.S. Department of Defense. Many forts and fortifications throughout history have also used a pentagonal layout for defensive advantages.
Nature
In nature, pentagonal symmetry can be seen in some flowers, like the morning glory, and in fruits like okra. Echinoderms, such as starfish, often exhibit fivefold symmetry.
Design and Art
The shape is used in logos, patterns, and artistic designs. The panels of a traditional soccer ball are made of pentagons and hexagons.

Examples in the Real World

  • The Pentagon building in the USA.
  • Home plates in baseball are irregular pentagons.
  • The Chrysler logo.

Common Questions and Key Facts

  • Frequently asked questions about pentagons.
  • Correcting common misconceptions.
  • Interesting facts and trivia.
Can a pentagon tile a plane?
Regular pentagons cannot tile a plane by themselves without gaps. This is a unique property compared to triangles, squares, and hexagons. However, some irregular pentagons can.
What is the sum of interior angles in a pentagon?
The sum of the interior angles of any simple (non-self-intersecting) pentagon is always 540°. For a regular pentagon, each angle is 540° / 5 = 108°.
Is it possible to construct a regular pentagon with only a compass and straightedge?
Yes, the construction of a regular pentagon using only a compass and straightedge was proven by the ancient Greek mathematicians and is a classic geometric construction.

Did You Know?

  • The number of diagonals in a regular pentagon is 5.
  • Pentagonal numbers are part of the family of polygonal numbers.