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Pentagon Calculator

A pentagon is a five-sided polygon where all sides and angles are equal in a regular pentagon. Each interior angle measures 108°, and the five angles sum to 540°. Whether you're working on geometry homework, architecture, or design projects, this calculator instantly derives all key measurements—side length, perimeter, area, diagonal, height, circumradius, and inradius (apothem)—from a single input value.

Last updated:

Creators Jasmine J. Mah, BSc
1,538 people find this calculator helpful

Understanding the Pentagon

A pentagon is a five-sided polygon. In a regular pentagon, all five sides are equal in length and all interior angles are equal. Each interior angle measures exactly 108°. You can verify this by dividing the sum of interior angles (540°) by five sides: 540° ÷ 5 = 108°.

Regular pentagons appear throughout nature and design—from flower petals and starfish to architectural elements. The ratio of the diagonal to the side length of a regular pentagon equals the golden ratio (φ ≈ 1.618), a proportion prized by artists and architects.

A pentagon differs from a pentagram, which is a five-pointed star formed by extending the sides of a pentagon or connecting non-adjacent vertices.

Key Formulas for a Regular Pentagon

All measurements of a regular pentagon derive from its side length a. Use these formulas to calculate perimeter, area, diagonal, height, and both radii:

Perimeter = 5 × a

Area = a² × √(25 + 10√5) / 4

Diagonal = a × (1 + √5) / 2

Height = a × √(5 + 2√5) / 2

Circumradius (R) = a × √(50 + 10√5) / 10

Inradius (apothem) = a × √(25 + 10√5) / 10

  • a — Side length of the regular pentagon
  • Perimeter — Total distance around the pentagon
  • Area — Total surface enclosed by the pentagon
  • Diagonal — Straight line connecting two non-adjacent vertices
  • Height — Perpendicular distance from the base to the opposite vertex
  • Circumradius (R) — Radius of the circle passing through all five vertices
  • Inradius (apothem) — Radius of the largest circle that fits inside the pentagon

How to Use the Pentagon Calculator

The calculator works by entering any single measurement—side length, perimeter, area, diagonal, height, or either radius. The tool then computes all remaining properties automatically.

Example: If you know the side length is 5 cm, enter it and the calculator will return:

  • Perimeter: 25 cm
  • Area: ≈ 43.01 cm²
  • Diagonal: ≈ 8.09 cm
  • Height: ≈ 6.88 cm
  • Circumradius: ≈ 4.25 cm
  • Inradius: ≈ 3.44 cm

This is particularly useful when you know the pentagon's area or perimeter but need to find the side length and other dimensions for construction or design work.

Common Pitfalls and Considerations

Keep these points in mind when working with pentagon calculations:

  1. Regular vs. irregular pentagons — This calculator assumes a <em>regular</em> pentagon where all sides and angles are identical. Real-world pentagons (like the Pentagon building) are rarely perfectly regular. The formulas will not apply to irregular pentagons.
  2. The golden ratio connection — The diagonal-to-side ratio of a regular pentagon equals the golden ratio (φ ≈ 1.618). This appears nowhere else in geometry and makes pentagons unique. Use this as a quick sanity check: diagonal should always be about 1.618 times the side length.
  3. Precision with radii — The circumradius (R) and inradius (r) involve nested square roots, which can accumulate rounding errors. If maximum accuracy matters, preserve more decimal places in intermediate steps rather than rounding early.
  4. Unit consistency — Ensure all your input values use the same units. If you enter the side in inches, the area will be in square inches, the perimeter in inches, and the diagonal in inches. Always check your units before trusting the output.

Angles in a Regular Pentagon

Every interior angle of a regular pentagon is 108°. This can be derived from the formula for interior angles of any polygon: (n − 2) × 180° / n, where n is the number of sides.

For a pentagon: (5 − 2) × 180° / 5 = 3 × 180° / 5 = 540° / 5 = 108°.

The exterior angle at each vertex is 180° − 108° = 72°. Five exterior angles sum to 360°, which is true for any convex polygon.

Frequently Asked Questions

What is the relationship between a pentagon's diagonal and its side?

The diagonal of a regular pentagon divided by its side length always equals the golden ratio, approximately 1.618. Mathematically, diagonal = side × (1 + √5) / 2. This unique ratio appears nowhere else in basic geometry and reflects the pentagon's special symmetry properties. Many artists and architects deliberately use this ratio in designs because it is aesthetically pleasing.

Can I find the area if I only know the perimeter?

Yes. Since perimeter = 5 × side, you can calculate the side length by dividing the perimeter by 5. Once you have the side, apply the area formula: area = a² × √(25 + 10√5) / 4. For example, a perimeter of 50 units gives a side of 10 units, and an area of approximately 172.05 square units.

What is the apothem of a pentagon?

The apothem (also called the inradius) is the perpendicular distance from the center of the pentagon to the midpoint of any side. It's the radius of the largest circle that fits entirely inside the pentagon. Calculate it with: apothem = a × √(25 + 10√5) / 10, where <em>a</em> is the side length. For a 10-unit side, the apothem is approximately 6.88 units.

How do the circumradius and inradius relate to each other?

The circumradius (R) is the radius of the circle passing through all five vertices, while the inradius (r) is the radius of the circle inscribed within the pentagon. Their ratio is constant: R / r = √5 / 2 ≈ 1.118. Both depend on the side length, but the inradius is always smaller because it must fit inside the pentagon without touching the sides' endpoints.

Why is the pentagon significant in geometry and nature?

The regular pentagon's connection to the golden ratio makes it mathematically elegant and frequently appears in nature—in flower petals, starfish, and hurricanes. Its 108° interior angles and five-fold rotational symmetry have captivated mathematicians and architects for millennia. The pentagon is also the foundation of the pentagram, a five-pointed star with deep historical and symbolic significance.

What's the difference between a pentagon and a pentagram?

A pentagon is a five-sided polygon with five vertices connected by straight edges. A pentagram is a five-pointed star, created either by extending the sides of a pentagon or by connecting every second vertex. Both share the same underlying geometry, but a pentagram has a hollow center and pointed arms, whereas a pentagon is a simple closed shape.

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