on April 26, 2022

How do you find the sides of an octagon with diameter?

Space & Navigation

Okay, so you want to figure out the side length of an octagon when all you know is its diameter? Sounds simple, right? Well, it can be, but there are a few twists depending on the type of octagon and what you actually mean by “diameter.” Let’s break it down, shall we? Think of this as your friendly guide to octagon side-sleuthing.

First things first: what is an octagon? It’s just a shape with eight sides. Now, if all those sides and angles are equal, we’ve got ourselves a regular octagon – nice and symmetrical. But if they’re all wonky and different lengths, that’s an irregular octagon. And that’s where things can get a bit hairy.

Then there’s the “diameter” question. It’s not as straightforward as you might think. We’re not just talking about circles here! When it comes to octagons, “diameter” could mean a few different things:

  • The Inscribed Circle’s Diameter (Apothem Connection): Imagine the biggest circle you could squeeze inside the octagon, just touching each side. That circle’s diameter is one option. Its radius is called the apothem, which will be useful later.

  • The Circumscribed Circle’s Diameter: Now picture a circle that goes around the octagon, hitting all eight corners. Its diameter is another possibility.

  • Width (Across the Flats): For a regular octagon, this is just the distance between two opposite, parallel sides. Think of it as measuring the octagon at its widest point. This is also the same as the diameter of the inscribed circle.

  • Vertex-to-Vertex Distance: This could be the longest diagonal or some other distance between corners.

    • Cracking the Code: Regular Octagons

    Let’s stick with the regular octagons for now – they’re much easier to deal with.

    1. Circumscribed Circle Diameter (D) is Known:

    • Here’s the magic formula: s = D * sin(π/8) or s = D * sin(22.5°). Basically, the side length (s) is related to the diameter (D) by a sine function. Who knew trigonometry could be so useful?

    • Quick cheat: sin(22.5°) is roughly 0.3827, so you can use s ≈ 0.3827 * D.

    • Example Time: Let’s say that outer circle has a diameter of 10 cm. Then, the side length is about 0.3827 * 10 cm = 3.827 cm. Not bad, eh?

    2. Inscribed Circle Diameter (d) or Width (Across Flats) is Known:

    • Different diameter, different formula: s = d * tan(π/8) or s = d * tan(22.5°). Now we’re using the tangent function.

    • Another shortcut: tan(22.5°) is about 0.4142, giving us s ≈ 0.4142 * d.

    • Example: If that inner circle has a diameter of 10 cm, then the side length is roughly 0.4142 * 10 cm = 4.142 cm.

    3. Got the Apothem (a)?

    • Remember the apothem? It’s half the diameter of the inscribed circle (a = d/2). So, if you know the apothem:

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