Pentagon Calculator
Enter any one measurement of a regular pentagon and get all the rest instantly: area, perimeter, side length, diagonal, height, apothem and circumradius. Switch the "solve from" dropdown to start from the value you already know, and flip the unit toggle between metric and imperial.
Formula
Worked example
Side a = 5 m: area = 1.7204774 x 25 = 43.0119 m^2, perimeter = 25 m, diagonal = 1.6180340 x 5 = 8.0902 m, height = 1.5388418 x 5 = 7.6942 m, apothem = 0.6881910 x 5 = 3.4410 m, circumradius = 0.8506508 x 5 = 4.2533 m.
What a regular pentagon is
A regular pentagon is a five-sided polygon where every side has the same length and every interior angle is 108 degrees. Because the interior angles of any pentagon sum to 540 degrees, and a regular one divides that equally, each angle is exactly 540 / 5 = 108 degrees. This high symmetry means one measurement, any one of the seven properties this calculator outputs, is enough to determine the entire figure completely.
How reverse-solve works
The "Solve from" dropdown lets you start from the measurement you actually have. If you know the area, the calculator inverts the area formula to find the side: a = sqrt(A / K) where K is approximately 1.7204774. From the diagonal, a = d / 1.6180340 (dividing by the golden ratio). From the height, a = h / 1.5388418. From the apothem, a = r / 0.6881910. From the circumradius, a = R / 0.8506508. Once the side is known, every other property follows from the standard linear or quadratic scale factors shown in the steps panel.
Area and why it scales with side squared
A regular pentagon can be split into five identical isosceles triangles meeting at the center. Each triangle has base a and height equal to the apothem r, so the total area is (1/2) * a * r * 5 = (5/2) * a * r. Substituting r = (a/2) * cot(36 degrees) gives the compact expression A = (1/4) * sqrt(5(5+2*sqrt(5))) * a^2 with the constant approximately 1.7204774. Because both a and r scale linearly with the side, area scales with the square, so doubling the pentagon side makes it four times as large.
Height vs. apothem vs. circumradius
Three different "radial" measurements appear in a pentagon. The apothem r is the shortest: it runs from the center perpendicular to a side (the inradius, about 0.6882 * a). The circumradius R is longer: it runs from the center to a corner (about 0.8507 * a). The height h is the total span from one vertex to the opposite side measured perpendicularly, which equals the apothem plus the circumradius, approximately 1.5388 * a. In practical terms, height is what you need when fitting a pentagon into a bounding rectangle, circumradius when drawing the circumscribed circle, and apothem when placing the inscribed circle or computing the area from triangular sections.
The golden ratio inside a pentagon
The diagonal of a regular pentagon (the line connecting two non-adjacent vertices) is exactly phi = (1 + sqrt(5)) / 2 = 1.6180... times the side length. This is not a coincidence: the diagonals of a regular pentagon intersect in the golden ratio, and the triangle formed by the diagonals and one side is a golden gnomon. The same ratio recurs in the Fibonacci sequence and appears throughout classical and Renaissance art as a proportion considered especially harmonious.
Regular pentagon: all properties as multiples of the side length
| Property | Formula | Constant (a = 1) | Interior angle |
|---|---|---|---|
| Area (A) | (1/4)*sqrt(5(5+2*sqrt(5))) * a^2 | 1.7204774 | 108 deg x 5 |
| Perimeter (P) | 5a | 5 | - |
| Diagonal (d) | phi * a | 1.6180340 | - |
| Height (h) | (apothem + circumradius) | 1.5388418 | - |
| Circumradius (R) | a / (2*sin36deg) | 0.8506508 | - |
| Apothem / inradius (r) | (a/2)*cot36deg | 0.6881910 | - |
| Interior angle | 108 deg each | - | 540 deg total |
Multiply the constant in the right column by your side length a (area uses a squared).
Frequently asked questions
How do I calculate the area of a regular pentagon?
Use A = (1/4) * sqrt(5(5+2*sqrt(5))) * a^2 where a is the side length. The constant is approximately 1.7204774, so for a side of 5 the area is 1.7204774 * 25 = 43.01 square units. Alternatively, enter the side in this calculator and read the area directly.
Can I calculate the side length if I only know the area or the diagonal?
Yes. Use the "Solve from" dropdown to tell the calculator which measurement you already have. For area: a = sqrt(A / 1.7204774). For diagonal: a = d / 1.6180340 (the golden ratio). For height: a = h / 1.5388418. For apothem: a = r / 0.6881910. For circumradius: a = R / 0.8506508. The calculator handles all of these reverse-calculations automatically.
What is the difference between apothem, height and circumradius in a pentagon?
The apothem (inradius) is the distance from the center to the middle of a side, about 0.6882 times the side. The circumradius is the distance from the center to a vertex, about 0.8507 times the side. The height is the full perpendicular span from a vertex to the opposite side, about 1.5388 times the side. Height equals apothem plus circumradius, and all three are proportional to the side length.
What is the interior angle of a regular pentagon?
Each interior angle is exactly 108 degrees. The five angles sum to 540 degrees, consistent with the general polygon formula (n - 2) * 180 degrees, which for n = 5 gives 3 * 180 = 540 degrees. Each exterior angle is 72 degrees, and all five exterior angles sum to 360 degrees.
Does this calculator work for irregular pentagons?
No. These formulas assume a regular pentagon: five equal sides and five equal 108-degree angles. An irregular pentagon has unequal sides or angles, so its area must be found by other methods such as dividing it into triangles using coordinates.
Why does the diagonal of a pentagon equal the golden ratio times the side?
This emerges directly from the geometry of a regular pentagon. The diagonals intersect each other in the golden ratio phi = (1 + sqrt(5)) / 2, and the ratio of the diagonal to the side is also phi. Algebraically, because the five vertices of a regular pentagon lie on a circle, the chord lengths satisfy the Ptolemy relation, which yields phi as the ratio of the long chord (diagonal) to the short chord (side).
Sources
- Weisstein, Eric W. "Regular Pentagon." MathWorld, Wolfram
- Coxeter, H. S. M. "Introduction to Geometry." Wiley, 1969. Sections on regular polygons and the golden section.