If you need diagonal across 5 sides or diagonal across 4 sides or diagonal across 3 sides and you already know side a and angle α and angle β and perimeter and area and circumcircle radius (r) and incircle radius / apothem (r), this calculator does the substitution for you. It is built for quick answers, for checking somebody else’s arithmetic, and for the moments when a formula book is not within reach.
What Is the Decagon Calculator?
It is built around diagonal across 5 sides, diagonal across 4 sides, diagonal across 3 sides, and it works the same way whether you are checking a homework answer, confirming a colleague's figure, or working something out for yourself. Everything runs in your browser, so nothing you enter is uploaded anywhere.
How the Decagon Calculation Works
Nothing is hidden: each output is worked out from the values above it, which makes it easy to see where a surprising answer came from.
The relationship between side a, angle α, angle β, perimeter, area, circumcircle radius (r), incircle radius / apothem (r) and diagonal across 5 sides, diagonal across 4 sides, diagonal across 3 sides, diagonal across 2 sides, number of sides is fixed, so the same figures always give the same answer. That makes it a good cross-check: run the numbers a second way and see whether the result agrees.
Getting a Result You Can Trust
It is worth running the figures twice with slightly different inputs. A calculator that behaves sensibly will move its outputs by a proportionate amount; if a small change produces a wildly different answer, there is usually a unit or a decimal point issue somewhere in the inputs.
Frequently Asked Questions
What do I need to work out diagonal across 5 sides?
Enter side a, angle α, angle β, perimeter, area, circumcircle radius (r), incircle radius / apothem (r). The calculator then works through the relationship and reports diagonal across 5 sides, diagonal across 4 sides, diagonal across 3 sides.
What does the decagon actually calculate?
It takes side a and angle α and angle β and perimeter and area and circumcircle radius (r) and incircle radius / apothem (r) and converts them into diagonal across 5 sides, diagonal across 4 sides, diagonal across 3 sides using the standard relationship for this problem. Each step is applied in turn, so every output is traceable back to the numbers you typed in.
How should I enter side a?
Use the value you actually have, in the unit shown next to the field. Enter every value in the same unit system. The calculator assumes the figures you supply are consistent, so it cannot correct a mistake that happens before the calculation starts.
Why does a small change in one value move the answer so much?
That happens when an output depends on a ratio, because dividing by a small number magnifies any change in it. Check the units first, then check whether the value you entered is in the range the relationship was designed for.
Can I use this for a quick estimate as well as a precise answer?
Yes. The calculator is built for quick checks, so round your inputs to a sensible number of significant figures and the result will be just as useful. If you need more precision, keep the same method and add digits to the inputs rather than adjusting the method itself.