Pentagon Calculator

What Is a Pentagon?

A pentagon is a polygon with five sides and five angles. When all sides and angles are equal, it's called a regular pentagon. In a regular pentagon:

• All five sides are equal length
• Every interior angle is 108°
• The sum of all interior angles is 540°
• It can be divided into 5 isosceles triangles

Our calculator computes every property of a regular pentagon from any single known value — enter the side length, area, perimeter, diagonal, circumradius, or apothem, and all other properties are calculated automatically.

Pentagon Properties & Formulas

Area of a Pentagon

The area formula for a regular pentagon involves the square root of 5:

A = (a²/4) × √(5(5 + 2√5))

This simplifies to approximately A ≈ 1.72048 × a².

Example: For a pentagon with side length 5:

• A = (25/4) × √(5 × 9.4721) = 6.25 × √47.3607 = 6.25 × 6.882 ≈ 43.01

Perimeter of a Pentagon

Since all five sides are equal:

P = 5a

For a = 5: P = 5 × 5 = 25

Diagonal of a Pentagon

The diagonal connects two non-adjacent vertices. In a regular pentagon, the diagonal-to-side ratio is the famous golden ratio φ (phi):

d = a × φ = a × (1 + √5)/2 ≈ 1.618 × a

For a = 5: d = 5 × 1.618 ≈ 8.09

Each pentagon has 5 diagonals, all of equal length in a regular pentagon.

Height of a Pentagon

The height is the perpendicular distance from one side to the opposite vertex:

h = a × √(5 + 2√5) / 2 ≈ 1.539 × a

For a = 5: h ≈ 5 × 1.539 ≈ 7.69

Circumradius (R)

The circumradius is the radius of the circle that passes through all five vertices (circumscribed circle):

R = a × √(50 + 10√5) / 10 ≈ 0.851 × a

For a = 5: R ≈ 4.253

Apothem / Inradius (r)

The apothem (or inradius) is the perpendicular distance from the center to the midpoint of a side (inscribed circle radius):

r = a × √(25 + 10√5) / 10 ≈ 0.688 × a

For a = 5: r ≈ 3.441

The Golden Ratio & Pentagons

The regular pentagon has a deep connection to the golden ratio φ ≈ 1.61803. The ratio of the diagonal to the side length equals φ exactly. This makes pentagons a fundamental shape in both mathematics and art.

The golden ratio also appears in: the Fibonacci sequence, the Parthenon's proportions, sunflower seed spirals, and the famous pentagram (five-pointed star drawn inside a pentagon).

Frequently Asked Questions

What is a regular vs. irregular pentagon?

A regular pentagon has all five sides equal and all five angles equal (108° each). An irregular pentagon has sides and angles of different lengths and measures. Our calculator handles regular pentagons only, since irregular pentagons require all five side lengths and additional information to solve.

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. For a pentagon: 5(5−3)/2 = 5. In a regular pentagon, all diagonals are the same length.

Why is the interior angle 108°?

The sum of interior angles of any polygon is (n−2) × 180°. For a pentagon: (5−2) × 180° = 540°. Dividing equally among 5 angles: 540° / 5 = 108°.

What is the difference between circumradius and apothem?

The circumradius (R) is the distance from the center to a vertex (corner). The apothem (r) is the distance from the center to the midpoint of a side. The circumradius is always larger: R ≈ 1.236 × r.

Where are pentagons found in real life?

The most famous example is the Pentagon building in Washington, D.C. Pentagons also appear in soccer ball patterns, certain crystals, starfish body plans, and many tiling patterns. The pentagram (five-pointed star) inscribed in a pentagon is an ancient mathematical symbol.