Pentagon calculator
A regular pentagon with sides of 2 has a perimeter of 10 and an area of about 6.8819. Enter the side length to get both measures together with the formula behind them. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Side
- 2
- Area
- 6.8819096024
- Perimeter
- 10
A regular pentagon has five equal sides and five equal interior angles of one hundred eight degrees. Builders, teachers, game artists, landscapers, and fabricators often know only the edge length of a pentagonal tile, plaque, home-plate outline, gazebo floor, or decorative panel and need both the enclosed area and the outer perimeter in one deterministic step. This regular pentagon calculator accepts a single positive finite side length, applies the closed forms for area and perimeter, and returns both values together with a short formula string you can show next to the answer.
What a regular pentagon is and how area and perimeter follow from one side
A regular pentagon is the unique convex five-sided polygon whose sides are all equal and whose interior angles are all equal, each measuring one hundred eight degrees. Connecting the centre to the five vertices partitions the figure into five congruent isosceles triangles with vertex angle seventy-two degrees; the area of the pentagon is five times the area of one of those triangles, and the perimeter is simply five times the common side length s.
The closed-form area that follows from that decomposition is A equals one quarter times the square root of five times the quantity five plus two square-root-of-five, times s squared: A = (1/4)·√(5(5+2√5))·s². An equivalent trigonometric statement is A equals (5/4)·s²·cot(π/5), or A equals (5·s²) divided by (4·tan(π/5)); all three identities are algebraically the same for a perfect regular pentagon. The implementation uses the nested-radical form so a single deterministic floating-point path never depends on a library tangent of π/5.
Where a regular pentagon calculator earns its keep
Architecture and interior design use regular pentagonal tiles, windows, and floor plates; knowing area from the published edge length sizes material orders, while the perimeter sizes trim, gasket, or edge banding. Manufacturing of pentagonal plaques, medals, and sheet-metal blanks needs the same pair of numbers when quoting stock and when nesting parts on a sheet.
Game and map engines that place pentagonal rooms, shields, or home-plate collision hulls often store side length as the canonical size and must convert to area for paint, lighting, or physics density and to perimeter for outline shaders. Landscaping a five-sided bed or gazebo floor follows the same arithmetic: mulch volume starts from area, edging length starts from perimeter.
Keep requests small, pin the rounded area and perimeter in tests, and you get reproducible plane measures for homework, LMS plugins, CAD macros, and edge workers without standing up a general polygon library when the only figure that matters is the regular pentagon.
What you can do with it
Pentagonal tile and plaque fabrication
Convert a published pentagon edge length into area for stock and coating, and into perimeter for trim, gasket, or edge banding.
Game, CAD, and landscape layout
Derive fill area and outline length from a side-based pentagon size for maps, collision hulls, gazebo floors, and five-sided beds.
FAQ
What happens if the side length is zero or negative?
Side length must be strictly greater than zero; non-positive values are not billed.
Is this the same as a pentagonal prism or a pentagonal number?
No. This tool measures a regular plane pentagon. A pentagonal prism is a solid, and a pentagonal number is the figurate sequence P(n)=n(3n−1)/2.
Which units does the calculator use?
None are assumed. Area uses the square of the input unit and perimeter uses the same linear unit as side—millimetres in, mm² and mm out.
Does it work for an irregular pentagon?
No. The formulas only hold for a regular pentagon, with five equal sides and five equal angles of 108°. For a five-sided figure with unequal sides the result would be wrong, so measure that shape another way.
Is there another way to write the pentagon area formula?
Yes. The area (1/4)·√(5(5+2√5))·s² is the same as 5s² divided by 4·tan(π/5). The calculator uses the version with square roots, so its answer does not depend on how a tangent is computed.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/math/pentagon \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"side":2}'const res = await fetch("https://api.kit.forhosting.com/math/pentagon", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"side": 2
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/math/pentagon",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"side": 2
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/math/pentagon", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"side":2}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"side":2}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/math/pentagon", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"side": 2
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "math.pentagon",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
Non-positive or non-finite side values fail with invalid_input instead of a silent NaN.
What a regular pentagon is and how area and perimeter follow from one side
The perimeter needs no roots at all: P = 5s. This landing is not a general n-gon solver and is not a pentagonal-prism or pentagonal-number tool; people who search for a pentagon calculator want the two plane measures of the regular five-sided figure from the one number they already have.
Input contract, response shape, and validation rules
Zero and negative side lengths also fail, because a geometric edge length must be strictly greater than zero—there is no meaningful area or perimeter for a collapsed or inverted pentagon. Example: side equals two yields perimeter exactly ten and area equal to the square root of five times five plus two square-root-of-five, about six point eight eight one nine. Side equals one yields perimeter five and area about one point seven two zero five. Units are not converted: if you pass centimetres you get square centimetres for area and centimetres for perimeter; if you pass inches you get square inches and inches. POST a JSON body with one required field: side, a finite positive number giving the edge length of the regular pentagon. The value may arrive as a JSON number or as a numeric string; empty strings, missing fields, non-numeric text, NaN, and Infinity fail with invalid_input naming the field. On success the body echoes side, returns area and perimeter rounded to twelve decimal places for stable goldens across runtimes, and includes a formula string restating A = (1/4)·√(5(5+2√5))·s² and P = 5s. Failed validation does not consume prepaid balance; successful parse-tier calls use the published $0.002 base price.
Integration use cases
Classroom and exam checks
Let students enter side length s and verify A=(1/4)·√(5(5+2√5))·s² and P=5s against textbook methods without calculator slip-ups.
Technical questions
How do I find the area and perimeter of a regular pentagon from the side length?
Send a finite positive number side. The API returns area A = (1/4)·√(5(5+2√5))·s² and perimeter P = 5s, plus a short formula string.
What does the pentagon calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.