What this calculator does
A regular pentagon shape has five equal sides and five equal interior angles, each measuring 108°. Like any regular polygon, the whole figure is fixed by a single measurement, but that measurement might be given as the side length itself, the apothem (the distance from the centre to the middle of a side), or the circumradius (the distance from the centre to a corner).
This pentagon shape calculator accepts whichever of those three you already have, converts it to the side length internally, and returns the area, perimeter and the other two radii together, so the same figure never has to be worked out twice.
The formula
From the side length s, area = (5/4) × s² × cot(π/5). If the apothem is known instead, the side length is recovered first as s = apothem × 2 × tan(π/5); if the circumradius is known, s = circumradius × 2 × sin(π/5). Every other figure then follows from that side length.
| Term | Meaning |
|---|---|
| Side length | The length of one of the five equal edges. |
| Apothem | The perpendicular distance from the centre to the midpoint of a side, also the inradius. |
| Circumradius | The distance from the centre to any corner, the radius of the circle passing through all five vertices. |
The inputs explained
| Field | What to enter |
|---|---|
| Value | Enter the side length, apothem or circumradius, whichever measurement you already have. |
| This value is the | Select which measurement the value above represents. |
When to use it
A pentagon shape from a known side length
A pentagon-shaped paving slab, sign or panel is usually specified by its edge length, the most direct route to the area and perimeter here.
Working from a measured apothem
A pentagon inscribed inside a circular frame or fitted flush against a flat gap is easier to measure across its apothem, the flat-to-flat style distance, than along a single sloped edge.
Working from a circumradius or bounding circle
A pentagon designed to fit inside a given circle, such as a badge, coin face or bolt pattern, starts from the circumradius, the radius of that bounding circle, rather than the side length.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
Pentagon area across a range of side lengths
The area and perimeter of a regular pentagon across a range of side lengths.
Does it matter whether the value 10 is the side, apothem or circumradius?
The number 10 entered once as a side length, once as an apothem and once as a circumradius, to show how the pentagon shape calculator’s answer changes with what that number represents.
| The value 10 is the | Area | Side length |
|---|---|---|
| Side length | 172.048 | 10.000 |
| Apothem | 363.271 | 14.531 |
| Circumradius | 237.764 | 11.756 |
Questions
What is a regular pentagon shape?
A regular pentagon is a five-sided polygon with all five sides the same length and all five interior angles equal, each measuring 108°.
What is the area formula for a pentagon shape?
Area = (5/4) × s² × cot(π/5), where s is the side length. This works out to roughly 1.72 times the square of the side length.
What is the interior angle of a pentagon?
Each interior angle of a regular pentagon is 108°. The five interior angles always add up to 540°, following the general (n−2) × 180° rule for an n-sided polygon.
What is the difference between the apothem and the circumradius?
The apothem runs from the centre to the midpoint of a side, perpendicular to that side. The circumradius runs from the centre to a corner instead, and is always the longer of the two measurements for the same pentagon.
For a hexagon instead, see the hexagon calculator. For any number of sides at once, see the general regular polygon calculator.