Pentagon Calculator

Calculate all properties of a regular pentagon including area, perimeter, apothem, circumradius, and diagonal. Perfect for geometry, architecture, and design projects.

Length of one side of the regular pentagon

Area = (1/4)sqrt(25 + 10sqrt5) * s^2 ≈ 1.72048s^2 Perimeter = 5s Apothem = s/(2 * tan(36deg)) ≈ 0.688s Circumradius = s/(2 * sin(36deg)) ≈ 0.851s Diagonal = rho� * s ≈ 1.618s (golden ratio) Interior Angle = 108deg Exterior Angle = 72deg Where: s = side length, rho� = golden ratio
Example 1 (Tile Design): Side Length = 10 cm Area = 1.72048 * 10^2 = 172.05 cm^2 Perimeter = 5 * 10 = 50 cm Apothem = 0.688 * 10 = 6.88 cm Circumradius = 0.851 * 10 = 8.51 cm Diagonal = 1.618 * 10 = 16.18 cm Example 2 (Baseball Home Plate): Side Length = 12 in Area = 1.72048 * 12^2 = 247.75 in^2 Perimeter = 5 * 12 = 60 in Apothem = 0.688 * 12 = 8.26 in Circumradius = 0.851 * 12 = 10.21 in

What is a regular pentagon?

A regular pentagon is a five-sided polygon where all sides are equal in length and all interior angles are equal (108deg). The Pentagon building in Washington D.C. is a famous example of this shape.

What is the formula for the area of a regular pentagon?

The area of a regular pentagon is calculated as: Area = (1/4)sqrt(25 + 10sqrt5) * s^2, where s is the side length. This simplifies to approximately Area ≈ 1.72048 * s^2. For a pentagon with side 10 cm, area ≈ 1.72048 * 100 = 172.05 cm^2.

How do I calculate the perimeter of a pentagon?

For a regular pentagon, the perimeter is simply: Perimeter = 5 * side length. For example, if each side is 8 cm, the perimeter is 5 * 8 = 40 cm. For irregular pentagons, you would add all five individual side lengths.

What is the apothem of a pentagon?

The apothem is the distance from the center of the pentagon to the midpoint of any side. It is perpendicular to that side. For a regular pentagon: Apothem = s/(2 * tan(36deg)) ≈ 0.688 * s, where s is the side length.

What is the circumradius of a pentagon?

The circumradius is the radius of the circle that passes through all five vertices of the pentagon. For a regular pentagon: Circumradius = s/(2 * sin(36deg)) ≈ 0.851 * s, where s is the side length.

What are the interior and exterior angles of a regular pentagon?

Each interior angle of a regular pentagon is 108deg. Each exterior angle is 72deg. The sum of all interior angles is (5-2) * 180deg = 540deg. These angles are what give the pentagon its characteristic shape.

How many diagonals does a pentagon have?

A pentagon has 5 diagonals. The formula for the number of diagonals in any polygon is: n(n-3)/2, where n is the number of sides. For a pentagon: 5(5-3)/2 = 5 diagonals. Each diagonal connects non-adjacent vertices.

What are real-world applications of pentagon calculations?

Pentagon calculations are used in architecture (like the Pentagon building), design of home plates in baseball, decorative tile patterns, nuts and bolts in engineering, flower petal arrangements, crystal structures, and various geometric art and craft projects.

Can I calculate pentagon properties from the apothem?

Yes! If you know the apothem (a), you can find the side length: s = 2a * tan(36deg) ≈ 1.453 * a. Then calculate area using: Area = (5/2) * apothem * side length, which is often easier than the standard formula.

What is the golden ratio connection to pentagons?

Regular pentagons have a special connection to the golden ratio (rho� ≈ 1.618). The ratio of a diagonal to a side length in a regular pentagon equals the golden ratio. This relationship appears frequently in nature and art.