The Problem
Cut an octagon from a board. Sounds easy enough, but without the right measurements the sides will not all be the same size. (Said from experience.)![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgO5EFUlmMq1Oly8Lsq5IJMJMkEliwVX1JS2rAPMP0hFdo9WdBS81d3GUwx6v0gW3IpYDVJK8APL_N5c1IttXzNFUlpcfqoY2w9_uChcOkRA9vgGUEU_GuxfXMGXmyp9Y0gXSYdBVael4Q/s320/octagon.jpg)
The Solution
Use a little high school trig to determine where to mark the board for 45° cuts with the miter saw.Let's start with the Pythagorean theorem:
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiglkAVMhE7Aee3Qx2Ji36zH8USysC42JDXLomuZ0F8fVjjh8fYyp-a6A8gh-NIoQHNy-zClpab_vh6FXHssfJ0Q1d91ZAspjXmhAYL5KhqOj0-5U3UZ_s62Khmlx1LVB_DRRNmw9FWJDM/s320/blog_eq_1.jpg)
Now let's plug in our values. In this case, both legs of our right triangle are the same length (x).
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgU88oX0v6bIDqCDlCUAxARZDPy45lsprR0mZh6uXC-Mx5SyWspkb00qDNhUbmrwdxkVdb-JUeV5Dy_3tf7k3GW_IpeaLahT1U58sTssto-DFDJciL7l1L_tPP0NS61EtU9bYSOkpPx1X4/s320/blog_eq_2.jpg)
So now if we solve for y, we can determine x and vice versa. With a board of width w:
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEieUQhK0sqjd_AOq6xKbs1hxvlTUUaWjKb1ArNvEAr8pV0pwj2A4PJFvPCOed0-bsaBk1Ec0X0usCdvNCNHZ0Ko4br4AOD2343qEeUb-xPGgGGQPM3sERaw0Qy0cNcQRW5yFmpF6E5OXHI/s320/blog_eq_3.jpg)
Now we can just plug in our nominal lumber size w and get values for x and y. It helps to either have a project calculator that will return the results in fractions or a tape measure with decimal values.
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