Pentagon Calculator — area, perimeter, apothem and diagonal

The pentagon calculator above solves a regular five-sided figure from any one of five common measurements. The pentagon is the first regular polygon where the useful dimensions stop being obvious: it has no pair of parallel sides, so there is no single "width across", and its height and its width are two different numbers. Getting those straight is most of what this page is for.

Arb Digital publishes this free alongside the rest of our plane-geometry set. If your five-sided shape does not have equal sides and equal angles, this is the wrong tool — take the corner coordinates to our polygon calculator instead, which handles any outline.

What This Pentagon Calculator Does

It reports the area, perimeter, apothem, circumradius, diagonal and overall height of a regular pentagon — five equal sides, five equal 108° angles. Enter a side of 10 cm and you get an area of about 172.05 cm², a perimeter of 50 cm, an apothem of 6.882 cm, a diagonal of 16.180 cm and a height of 15.388 cm.

Notice that the diagonal and the height differ by about 5%. That is not a rounding artefact; it is a real property of odd-sided polygons, and it is covered in detail below. The quantity field multiplies the area for repeated facets or panels, which is useful when a pentagon is one face of a larger assembly.

How to Use It

1. Decide which dimension you have. A drawing usually gives the side; a physical measurement of a five-sided object usually gives the diagonal or the height.
2. Select that measurement type so the conversion to side length is handled for you.
3. Enter the value and pick a unit. Results come back in that unit, with area squared.
4. Add a quantity if you need the total across several identical pentagons.
5. Check the working line, which restates the conversion and the area calculation using your figures.

The Formula: How Pentagon Area Is Calculated

For a regular pentagon of side s, the area is A = ¼ √(5(5 + 2√5)) × s², which works out at approximately 1.720477 × s². With s = 10 cm, A = 1.720477 × 100 = 172.05 cm². The constant is ugly because the pentagon's internal geometry involves √5, which is also where the golden ratio comes from.

The general regular-polygon relation gives the same result more transparently: A = ½ × perimeter × apothem. The apothem of a regular pentagon is s / (2 tan 36°) ≈ 0.688191s. With a perimeter of 50 cm and an apothem of 6.88191 cm, the area is ½ × 50 × 6.88191 = 172.048 cm² — the same figure to three decimal places. MathWorld's pentagon entry gives the algebraic derivation of the closed form.

Why a Pentagon Has No Single Width

Hexagons and octagons have an even number of sides, so every side has a parallel partner directly opposite and "across flats" is a meaningful single number. A pentagon has five sides, so every side faces a vertex rather than another side. The consequence is that the shape has two distinct span measurements and neither is a diameter in the usual sense.

The diagonal joins two non-adjacent corners and equals φs, where φ is the golden ratio 1.618034. That is the widest the pentagon gets, so it is the number to use for clearance. The height runs from one vertex perpendicular to the opposite side and equals R + apothem = 1.538842s. Height is smaller than the diagonal, and both are larger than the side. Enter a diagonal into a field expecting a height, or vice versa, and your area will be out by about 10%.

Because a pentagon has no parallel faces, it also has a constant-width problem: rotate it inside a gap and the required gap changes. This is why five-sided fasteners exist as a security feature — a standard adjustable spanner grips a hex nut at any of three orientations, and a pentagon does not present flats in the same convenient way.

The Golden Ratio Lives Inside the Pentagon

The ratio of a regular pentagon's diagonal to its side is exactly the golden ratio, φ = (1 + √5)/2 ≈ 1.6180339887. This is not a coincidence or an approximation; it falls out of the 36-72-72 triangles the diagonals create. Draw all five diagonals and they form a pentagram, and the smaller pentagon at its centre is a scaled copy of the original, reduced by a factor of φ². That self-similarity is why the pentagon appears in so much classical proportion theory.

A practical use follows from it. If you can measure a diagonal accurately but the sides are hard to reach — a five-sided plot with obstructed edges, say — divide the diagonal by 1.618034 to get the side, then calculate everything else. This calculator does exactly that when you choose the diagonal input mode. MathWorld's page on the golden ratio covers the wider family of relationships.

Angles, Construction and Why Pentagons Do Not Tile

Each interior angle of a regular pentagon is 108°, each exterior angle is 72°, and the five interior angles sum to (5 − 2) × 180° = 540°. Three pentagons at a point give 324°, leaving a 36° gap; four give 432°, an overlap. So regular pentagons cannot tile a plane, which sets them apart from triangles, squares and hexagons. Irregular pentagons are a different story — fifteen distinct types of convex pentagon are known to tile the plane, and the classification of them was only completed in 2017.

For fabrication, the exterior angle of 72° means each mitre in a five-piece frame is 36°. That is a less convenient saw setting than the octagon's 22.5°, which is part of why octagonal and hexagonal frames are more common in joinery than pentagonal ones despite the pentagon needing fewer pieces.

Where Regular Pentagons Are Used

The pentagon appears wherever five-fold symmetry is wanted or unavoidable. A traditional football is a truncated icosahedron with twelve regular pentagonal panels and twenty hexagonal ones — the pentagons are what allow the flat sheet to curve into a ball, since a hexagon-only arrangement stays flat. The same principle drives geodesic domes, which always contain exactly twelve pentagonal vertices no matter how finely subdivided they are.

Elsewhere, pentagons show up in home plate on a baseball diamond, in five-sided security fasteners used on utility covers, in some paving and tile patterns, and in crystallography where five-fold symmetry was long thought impossible until quasicrystals were discovered. In each case the calculation you need is the same: get to the side length, then apply the constant.

Splitting a Pentagon Into Triangles

If you would rather not memorise the 1.720477 constant, there are two decompositions that get you to the same answer with nothing but a triangle formula, and both are worth knowing because they generalise to any regular polygon.

The first splits the pentagon from its centre into five identical isosceles triangles, each with base s and height equal to the apothem. Each has area ½ × s × apothem, and five of them give (5/2) × s × 0.688191s = 1.720477s². That is where the constant comes from, and it is the same reasoning behind the universal ½ × perimeter × apothem rule.

The second splits the pentagon from one vertex into three triangles by drawing two diagonals. This is the general fact that any n-sided polygon divides into n − 2 triangles from a single vertex, which is also why the interior angles sum to (n − 2) × 180°. It is less convenient for a regular pentagon because the three triangles are not identical, but it is the method that still works when the shape is irregular — and it is essentially what the shoelace formula automates.

Both routes are worth checking your answer against. If a decomposition and the closed formula disagree, the usual cause is that the shape being measured is not actually regular.

Comparing the Pentagon With Its Neighbours

For a fixed side length of 10, an equilateral triangle has an area of about 43.30, a square 100, a pentagon 172.05, a hexagon 259.81 and an octagon 482.84. Area grows faster than the side count because each extra side adds both length and interior room.

For a fixed perimeter of 50, the comparison inverts into something more useful: a square encloses 156.25, a pentagon 172.05, a hexagon about 180.42 and a circle about 198.94. More sides always enclose more area for the same boundary, with the circle as the ceiling. A pentagon captures about 86.5% of the circle's area for the same perimeter. Our hexagon calculator, octagon calculator and circle calculator cover those cases, and the perimeter calculator compares boundaries across shapes directly.

Common Mistakes to Avoid

• Using the diagonal as the side — the diagonal is 1.618 side lengths, so this inflates the area by a factor of about 2.6.
• Confusing height with width — a regular pentagon's height (1.5388s) and its widest span (1.6180s) are different numbers, unlike in an even-sided polygon.
• Applying the regular formula to an irregular pentagon — five sides is not enough; all five sides and angles must be equal.
• Assuming pentagons tile — regular ones do not, because 108° does not divide into 360°.
• Cutting frame mitres at 72° — the exterior angle is 72°, so each of the two mitred ends takes half of it, at 36°.

Related Free Tools From Arb Digital

Handle any number of sides, regular or irregular, with the polygon calculator, work in rectangles with the rectangle area calculator, split a shape into three-sided pieces with the triangle area calculator, convert dimensions with the length converter, and convert a finished area with the area converter. Everything is listed in our free online tools hub.

Frequently Asked Questions