Regular Pentagon Calculator – Area, Perimeter, Circumradius & Inradius

Pentagon Geometry

Side length (s)

The Regular Pentagon Calculator calculates the perimeter, area, circumradius (R), and inradius (r / apothem) of any regular five-sided polygon from a given side length. A regular pentagon is renowned for its fivefold rotational symmetry and its deep connection to the golden ratio (φ ≈ 1.618034). Enter your side length above for instant, highly accurate results with full step-by-step algebraic working.

Key Regular Pentagon Formulas

Perimeter (P): P = 5 × s

Area (A): A = (1/4) × √(5(5 + 2√5)) × s² ≈ 1.720477 × s²

Circumradius (R): R = s / (2·sin(36°)) ≈ 0.850651 × s

Inradius / Apothem (r): r = s / (2·tan(36°)) ≈ 0.688191 × s

Diagonal Length (d): d = φ × s = ((1 + √5) / 2) × s ≈ 1.618034 × s

Real-World Applications of Pentagons

Iconic Architecture: The Pentagon headquarters of the United States Department of Defense is constructed as a massive concentric regular pentagon.
Botanical and Marine Symmetry: Countless flowers (such as morning glories, hibiscus, and okra blossoms) and marine creatures (starfish, sea urchins) exhibit natural fivefold pentamerous symmetry.
Graphic Design & Heraldry: The pentagram (5-pointed star formed by diagonals), national flags, company emblems, and medals utilize pentagonal proportions.
Fullerene Chemistry: Carbon-60 fullerene molecules (buckyballs) and virus capsids are composed of 12 regular pentagonal faces interlinked with hexagons.

The Golden Ratio (φ) and the Pentagon

In any regular pentagon, the ratio of a diagonal length to a side length is strictly equal to the golden ratio: φ = (1 + √5) / 2 ≈ 1.618034. When all five diagonals of a pentagon are drawn, they intersect at golden sections and form a smaller, inverted regular pentagon at the centre.

The exact trigonometric values for the internal angles are directly tied to φ: sin(18°) = (√5 - 1) / 4 and cos(36°) = φ / 2. This makes the pentagon one of the most mathematically rich shapes in classical Euclidean geometry.

Mathematical Derivation of the Pentagon Area Formula

A regular pentagon can be decomposed into 5 congruent isosceles triangles radiating from its geometric centre. The central angle of each triangle is 360° / 5 = 72°. Bisecting this central angle creates two right triangles with a top angle of 36° and a base of s / 2.

The height of each triangle is the apothem (inradius r):

r = (s / 2) / tan(36°) = (s / 2) × √(5 + 2√5)

The area of one central triangle is (1/2) × base × height = (1/2) × s × r. Multiplying by 5 for all five triangles yields:

Area = 5 × (1/2 × s × r) = 2.5 × s × r = (1/4) × √(5(5 + 2√5)) × s² ≈ 1.720477 × s²

Interior & Exterior Angles of a Pentagon

For any n-sided polygon, the sum of interior angles is given by (n - 2) × 180°. For a pentagon (n = 5):

Sum of Interior Angles: (5 - 2) × 180° = 3 × 180° = 540°
Each Interior Angle (Regular): 540° / 5 = 108°
Each Exterior Angle: 360° / 5 = 72° (since 108° + 72° = 180°)
Number of Diagonals: n(n - 3) / 2 = 5(5 - 3) / 2 = 5 diagonals

Reference Table for Common Side Lengths

Side (s)	Perimeter (P)	Area (A)	Inradius (r)	Circumradius (R)	Diagonal (d)
1	5.0000	1.7205	0.6882	0.8507	1.6180
2	10.0000	6.8819	1.3764	1.7013	3.2361
3	15.0000	15.4843	2.0646	2.5520	4.8541
4	20.0000	27.5276	2.7528	3.4026	6.4721
5	25.0000	43.0119	3.4409	4.2533	8.0902
6	30.0000	61.9372	4.1291	5.1039	9.7082
8	40.0000	110.1105	5.5055	6.8052	12.9443
10	50.0000	172.0477	6.8819	8.5065	16.1803

Step-by-Step Manual Calculation Example

Given: Regular pentagon with side length s = 5 cm

1. Perimeter: P = 5 × 5 = 25 cm

2. Exact Area: A = (1/4) × √(5(5 + 2√5)) × 5² = (1/4) × √(5(5 + 4.472136)) × 25 = (1/4) × √47.36068 × 25 ≈ 1.720477 × 25 = 43.0119 cm²

3. Circumradius (R): R = 5 / (2 × sin(36°)) = 5 / (2 × 0.587785) = 5 / 1.17557 ≈ 4.2533 cm

4. Inradius (r / Apothem): r = 5 / (2 × tan(36°)) = 5 / (2 × 0.726543) = 5 / 1.453085 ≈ 3.4409 cm

5. Diagonal (d): d = 1.618034 × 5 = 8.0902 cm

Common Mistakes When Working with Pentagons

Using the Full Central Angle (72°) instead of Half-Angle (36°): Trigonometric formulas for circumradius and inradius divide the central triangle in half, requiring θ/2 = 36°.
Assuming Circumradius Equals Side Length: This property is only true for a regular hexagon (R = s). For a pentagon, R ≈ 0.85065 × s.
Confusing Inradius (Apothem) with Circumradius: The inradius r reaches the midpoint of a flat edge, while the circumradius Rreaches the outer vertex (R > r).
Approximating √5 Prematurely: Always maintain full decimal precision during intermediate steps to prevent rounding drift in the final area.

Frequently Asked Questions about Regular Pentagons

What is a regular pentagon?

A regular pentagon is a 5-sided polygon with all equal sides and equal interior angles of 108°. Its central angle between adjacent vertices is 72°.

How do you calculate the area of a regular pentagon?

The exact area formula is A = (1/4) × √(5(5 + 2√5)) × s² ≈ 1.720477 × s², where s is the side length. Alternatively, you can use the apothem: Area = 2.5 × s × r.

What is the difference between circumradius (R) and inradius (r)?

The circumradius (R) is the distance from the centre to any vertex (R = s / (2·sin(36°)) ≈ 0.85065·s). The inradius (r or apothem) is the perpendicular distance from the centre to the midpoint of any side (r = s / (2·tan(36°)) ≈ 0.68819·s).

How is the golden ratio related to a regular pentagon?

The diagonal of a regular pentagon divides it in the golden ratio (φ = (1 + √5)/2 ≈ 1.618034). The ratio of any diagonal (d) to side (s) is exactly φ (d = φ·s).

How do you find the diagonal of a regular pentagon?

The diagonal length (d) is calculated as d = φ × s = ((1 + √5) / 2) × s ≈ 1.618034 × s, where s is the side length.

What is the apothem of a regular pentagon?

The apothem (also called the inradius r) is the line segment from the centre of the pentagon perpendicular to one of its sides. For side s, apothem r = s / (2·tan(36°)) ≈ 0.688191 × s.

What is the sum of interior angles of a pentagon?

For any 5-sided polygon, the sum of interior angles is (n - 2) × 180° = (5 - 2) × 180° = 540°. In a regular pentagon, each interior angle is 540° / 5 = 108°.

Can regular pentagons tile a plane without gaps?

No. Because each interior angle is 108° (which does not divide 360° evenly), regular pentagons alone cannot tile a 2D plane without gaps. However, they form beautiful aperiodic patterns such as Penrose tiling when combined with other shapes.