Pentagon Calculator

A = √(25 + 10√5) ÷ 4 × s² P = 5s a = s ÷ (2·tan(π ÷ 5)) diagonal = φ·s interior angle = 108°

Side

The length of any one of the five sides, in centimetres. As with every regular polygon this is the only input there is: all five sides are equal and all five angles are fixed at 108 degrees, so the side alone determines the whole shape

Area

The space the pentagon covers, in square centimetres: the square root of twenty-five plus ten times the square root of five, divided by four, times the side squared. The constant is about 1.7205, which is the figure to carry around — a pentagon covers about 1.72 times the square drawn on one of its sides

Perimeter

The distance all the way around, in centimetres: five times the side, since all five are the same. It is the one output that is always exact whenever the side is, which makes it the figure to check the others against when a result looks wrong

Apothem

The distance from the centre straight out to the midpoint of a side, in centimetres, which is the side divided by twice the tangent of 36 degrees. It is the radius of the largest circle that fits inside the pentagon, and twice it is the width of the shape measured across its flats

Diagonal

The distance between two corners with one corner between them, in centimetres, which is the side times the golden ratio. There is only one diagonal on this page rather than the two a hexagon has, because a regular pentagon's five diagonals are all the same length — every one of them skips exactly one corner, so there is no longer and shorter pair to tell apart

Interior angle

The angle between two neighbouring sides, inside the pentagon: 108 degrees, fixed there by the side count alone. It does not depend on the side length, which is why this column of the reference table is the same number on every row. Half of it, 54 degrees, is the mitre a five-sided frame needs

φ

The golden ratio, about 1.6180339887, which is the number the diagonal is the side multiplied by. It is the ratio in which a pentagon's diagonals cut each other, and it is the reason this shape turns up in design and in plants far more often than its arithmetic would suggest

√5

The square root of five, about 2.2360680. It appears in the area through the expression twenty-five plus ten root five, and it is what makes the golden ratio an irrational number rather than a fraction — one plus the square root of five, over two

The page is for the pentagon as a physical object, and pentagons are usually decorative rather than structural, which changes what you want from the numbers. If you are cutting one out of a sheet — a table top, a sign, a paving stone, a garden bed — the area tells you what material you are using and the apothem tells you how to lay it out from the centre, since twice the apothem is the width across the flats and that is the measurement you can actually get a rule across. The diagonal is then what tells you whether the finished piece fits the space it is going into, because it is the widest line across the shape, corner to corner through the middle. If you are building something out of five pieces — a frame, a lantern, a gazebo roof, a planter — the 108-degree interior angle is the figure to reach for, since the mitre on each piece is half of it, 54 degrees, and that setting is the same for every regular pentagon whatever its size. The golden ratio is the reason to look at the diagonal column even when you do not need it: it is the shape's signature, and a pentagon whose diagonal is not 1.618 times its side is not a regular pentagon, whatever it looks like. And if you are comparing shapes rather than measuring one, the same-side table on this page and the two beside it give the fairest comparison there is: at a side of 6 the pentagon covers 61.9372, the hexagon 93.5307 and the octagon 173.8234, so the same edge buys more area the more sides there are, on the way to the circle.

1. A side of 6

   1. Perimeter: 5 × 6 = 30
   2. Diagonal: φ × 6 = 1.6180339… × 6 = 9.708203…, which rounds to 9.7082
   3. Apothem: 6 ÷ (2 × tan 36°) = 6 ÷ 1.453085… = 4.129145…, which rounds to 4.1291
   4. Area: 1.7204774… × 36 = 61.937187…, which rounds to 61.9372
   5. Interior angle: (5 − 2) × 180 ÷ 5 = 108, exact

   The input the page loads with. The check to run on it is the diagonal divided by the side: 9.7082 ÷ 6 is 1.618, which is the golden ratio, and that ratio holds on every row of the table and on every regular pentagon there has ever been. This is also the row that lines up with the two pages beside it at the same side length: a side of 6 gives 61.9372 here against 93.5307 on the hexagon page and 173.8234 on the octagon page — same edge, more sides, more area. The pentagon is the smallest of the three and also the widest for its area, since with only five sides it is the least circle-like of them.

2. A side of 1

   1. Perimeter: 5 × 1 = 5
   2. Diagonal: φ × 1 = 1.6180339…, which rounds to 1.618
   3. Apothem: 1 ÷ (2 × tan 36°) = 0.688190…, which rounds to 0.6882
   4. Area: 1.7204774… × 1 = 1.7204774…, which rounds to 1.7205
   5. Interior angle: 108

   The unit pentagon, and the row where the constants stand on their own: the area is 1.7205, which is the whole of the area constant, and the diagonal is 1.618, which is the golden ratio itself. Both are worth remembering in this form. Compare the area against the square of side one, whose area is 1, and against the triangle's 0.433 and the hexagon's 2.598: five sides lands between a square and a hexagon, which is the pattern the whole family follows.

3. A side of 3

   1. Perimeter: 5 × 3 = 15
   2. Diagonal: φ × 3 = 4.854101…, which rounds to 4.8541
   3. Apothem: 3 ÷ (2 × tan 36°) = 2.064572…, which rounds to 2.0646
   4. Area: 1.7204774… × 9 = 15.484296…, which rounds to 15.4843
   5. Interior angle: 108

   The row that shows the golden ratio at its most awkward and therefore most convincing: the diagonal is 4.8541 and the side is 3, and 4.8541 ÷ 3 is 1.618. A ratio that comes out to the same three digits on a side of 1, a side of 3 and a side of 6 is a property of the shape rather than of the arithmetic. This row also has the closest area and perimeter on the table — 15.4843 against 15 — which is a near miss rather than an equality, and the two are different quantities in different units anyway.

4. A side of 10

   1. Perimeter: 5 × 10 = 50
   2. Diagonal: φ × 10 = 16.180339…, which rounds to 16.1803
   3. Apothem: 10 ÷ (2 × tan 36°) = 6.881909…, which rounds to 6.8819
   4. Area: 1.7204774… × 100 = 172.047740…, which rounds to 172.0477
   5. Interior angle: 108

   A ten-centimetre pentagon, about the size of a paving stone. The apothem of 6.8819 is the figure to lay it out with: twice it, 13.7638, is the width across the flats. There is a coincidence in this row worth spotting before it looks like an error — the apothem here, 6.8819, is the same four digits as the area at a side of 2, further up the table. Both are the same square root expression reached from different directions, one of them a length and one of them an area, and the fact that they share four digits at these two sizes means nothing beyond that.

5. A side of 0

   1. Perimeter: 5 × 0 = 0
   2. Diagonal: φ × 0 = 0
   3. Apothem: 0 ÷ (2 × tan 36°) = 0
   4. Area: 1.7204774… × 0 = 0
   5. Interior angle: (5 − 2) × 180 ÷ 5 = 108

   Zero is a legal input rather than an empty box, and this row makes the interior angle's nature plainest: a pentagon of side zero has collapsed to a point, four of the five figures are zero, and the interior angle is still 108 degrees. It is a property of the shape and not of the size, so it outlives the size entirely. Leaving the box blank is the different case: with nothing in it the page shows no result at all, because there is nothing to work from.

The page is for regular pentagons only: all five sides the same length and all five angles 108 degrees. A pentagon that is not regular — a house-shaped one, or an elongated one — has no single side length and cannot be described by one number, and the page will quietly give you the regular pentagon with that side rather than telling you that the shape you have in mind is a different one. The outputs are in centimetres and square centimetres whatever the unit dropdown is set to, so a side entered in inches comes back as a centimetre answer you have to convert yourself. Four decimal places is a display width rather than a claim about precision, and it is generous for a measured side; the last two digits are not meaningful unless the input was exact. The interior angle is exact at 108 degrees and prints without decimals, but it is the same on every row, so a page that appears not to respond to a change of side has not frozen — four of the five figures moved. There is one diagonal on the page and not two, which is not an omission: a regular pentagon's five diagonals are all the same length, so there is no longer and shorter pair to print. The page is two-dimensional and has no field for material thickness, so it cannot give the volume of a pentagonal prism or the weight of a pentagonal plate. It also does not cover the five-pointed star drawn from the diagonals, which is a shape made of several pentagons' worth of triangles rather than one pentagon.

Why is the pentagon's diagonal always 1.618 times its side?

Because that number is the golden ratio, and a regular pentagon is one of the plainest places it appears. The ratio is defined as the one where the whole is to the larger part as the larger part is to the smaller, and a pentagon's diagonals cut each other in exactly that proportion — which makes the diagonal to the side the same ratio again. It holds on every pentagon of every size, which is why the diagonal column of the reference table can be checked by dividing any row's diagonal by its side.

Why does the page show only one diagonal when the hexagon page shows two?

Because a regular pentagon has only one diagonal length. Every diagonal joins two corners with exactly one corner between them, and by symmetry all five of them come out the same. A hexagon is different: with six corners there are two distinct spans, one that skips a corner and one that goes straight across, so that page prints both. Printing a long and a short diagonal for a pentagon would print the same number twice and invite you to look for a difference that is not there.

Why does the interior angle never change when I change the side?

Because a regular pentagon's angles are fixed by the number of sides and nothing else. The interior angle of a regular polygon is the side count minus two, times 180 degrees, divided by the side count; for five sides that is 108 degrees, and no side length appears anywhere in that expression. A pentagon of side 1 and a pentagon of side 100 have the same angles and differ only in size. It is why the last column of the reference table is the same number on every row, and it is what makes a mitre setting of 54 degrees work for every pentagon you will ever cut.

What is the apothem and why is it useful here?

It is the distance from the centre straight out to the middle of a side, and it is the radius of the largest circle that fits inside. For a pentagon it is the figure to lay the shape out with: twice it is the width across the flats, and the flats are easy to get a rule across while the corners are not. If you are marking out a pentagon on a sheet, measure the apothem out from the centre five times and you have the midpoints of all five sides.

Do regular pentagons tile a floor?

No, and this is the one shape in the family that will not. Three regular pentagons around a point leave a gap, because three times 108 degrees is 324 and a full turn is 360 — the missing 36 degrees is never filled. It is why pentagonal floors need a second shape to fill in, and it is the reason the pentagon is more often decorative than structural while the hexagon, which tiles perfectly, is the shape of tiles and honeycombs.

Can the side be zero, and is that the same as leaving the box empty?

Zero is a real input: four of the five outputs come back as zero, which is the honest answer for a pentagon with no size, and the interior angle is still 108 degrees because it does not depend on the size at all. A blank box is different — with nothing entered the page shows no result at all, because there is nothing to work from. That distinction matters while you are typing, since clearing the box to retype a number should not flash up a row of zeros.

• Pentagon — the five-sided polygon, with the area, the apothem, the diagonal and the 108-degree interior angle this page computes, and the golden-ratio proportion its diagonals cut each other in — Wolfram MathWorld (United States)
• Regular polygon — the family a regular pentagon belongs to, with the interior angle fixed by the side count and the apothem defined as the distance from the centre to a side — Wolfram MathWorld (United States)
• Apothem — the distance from the centre of a regular polygon to the midpoint of a side, which is the radius of the inscribed circle and, doubled, the width of this shape across its flats — Wolfram MathWorld (United States)
• 教育部关于印发义务教育课程方案和课程标准（2022 年版）的通知——The fifth item in the annex list of this notice is the Mathematics Curriculum Standards for Compulsory Education (2022 edition); the sides, perimeter and area of a regular polygon are part of the compulsory-education mathematics curriculum, and the wording of the standards and the grade-band breakdown are governed by that annex — 中华人民共和国教育部