Free Pentagon Calculator

The Regular Pentagon Calculator is a practical online tool that computes all essential geometric properties of a regular pentagon when any one known measurement is supplied. Whether you have the side length, the perimeter, the area, the diagonal, the height, or the circumcircle/incircle radius, simply enter that value and the remaining parameters are generated instantly. This makes it a versatile geometry calculator and a reliable polygon calculator for students, designers, and anyone working with five‑sided polygons.

What Defines a Regular Pentagon?

A regular pentagon is a five‑sided polygon in which all sides are equal in length and all interior angles are congruent. The interior angles sum to $540^{\circ}$; therefore each angle measures exactly $108^{\circ}$. Its symmetrical shape also means that all five vertices lie on a common circumcircle, and an inscribed circle touches each side at its midpoint. This type of simple regular pentagon is the subject of this pentagon properties calculator.

Core Formulas Used by the Calculator

The tool uses standard geometric relationships for a regular pentagon. Given the side length $a$, the following dimensions are computed:

• Perimeter $P = 5a$ — the total distance around the pentagon.
• Area $A = \dfrac{\sqrt{25 + 10\sqrt{5}}}{4} a^{2}$. If the circumcircle radius $R$ is known, the area can also be obtained as $A = \dfrac{5R^{2}}{4} \sqrt{\dfrac{5 + \sqrt{5}}{2}}$.
• Diagonal $D = \dfrac{1 + \sqrt{5}}{2} a$, which equals the golden ratio $\varphi \approx 1.618$ multiplied by the side length. A regular pentagon has five diagonals, all of the same length, that intersect to form a pentagram.
• Height $h = \dfrac{\sqrt{5 + 2\sqrt{5}}}{2} a$ — the perpendicular distance between two parallel sides (or from a vertex to the opposite side, depending on orientation).

In addition to these, the calculator returns the circumradius (radius of the circle passing through all five vertices) and the inradius (radius of the inscribed circle that touches each side). These values are derived from the input using trigonometric relations based on the internal angle of $108^{\circ}$.

Using the Calculator

Operation is straightforward. Pick any one known dimension of the regular pentagon and enter it into the corresponding field. The calculator instantly displays the remaining six properties. Input and output values preserve the same unit system: if you enter a side in feet, the perimeter and height will also be in feet, and the area will be in square feet. No manual unit conversion is needed, though all inputs should belong to the same unit for consistent results.

A Real‑World Example

The Pentagon building in Arlington, Virginia, offers a practical illustration. Public records indicate its width (the “height” of the pentagram) is about 1,414 feet. By entering this value as the height into the calculator, the tool returns the following approximate dimensions:

• Side length: 918.9 ft
• Diagonal: 1,486.8 ft
• Perimeter: 4,594 ft (roughly 0.87 mi)
• Area: 33.35 acres
• Circumcircle radius: 781.6 ft
• Incircle radius: 632.4 ft

These numbers closely match the actual measurements of the building, confirming the calculator’s accuracy.

Extending to Other Regular Polygons

Although this tool specializes in pentagons, the same approach — entering one known quantity to derive all others — applies to any regular polygon, including triangles, squares, hexagons, and octagons. Each regular shape has its own formulas linking side length, area, perimeter, and radii, but the workflow remains identical.