- Status
It depends on if you want the eight sided shape inset in the circle or completely containing the circle. I am going to assume the latter. If this is the case the circle is tangent to the sides at the center of each side. The triangle formed between the center of the circle, the tangent of a side and the end point of one of the sides is a right triangle and has an included angle of 22.5 degrees (360/16). Here is a picture:
For the short side of the triangle:
l = r * cos(theta) (where l == the short side, theta is the included angle and r is the distance from the center of the circle to the intersection of two sides) So:
4 = r * cos(22.5)
r = 4/cos(22.5)
For the short side:
h = r * sin( theta )
h = 4*sin(22.5)/cos(22.5) = 4* tan(22.5) = ~1.66
h is 1/2 the distance of each side. So each side should be: ~3.32 feet.
For the short side of the triangle:
l = r * cos(theta) (where l == the short side, theta is the included angle and r is the distance from the center of the circle to the intersection of two sides) So:
4 = r * cos(22.5)
r = 4/cos(22.5)
For the short side:
h = r * sin( theta )
h = 4*sin(22.5)/cos(22.5) = 4* tan(22.5) = ~1.66
h is 1/2 the distance of each side. So each side should be: ~3.32 feet.
Last edited:
eyekode is right on, but here's a site to bookmark for future use if you'd like.
http://rechneronline.de/pi/octagon.php
http://rechneronline.de/pi/octagon.php
We are not talking about exact measurements here. I do it by country boy figures. The diameter is 8'. The circumference is 8' times 3.1416. Country boys use 3. 8x3=24. for 8 sides, that means 24 divided by 8 which = 3'. I would look at the available material and make the side use the most efficient use of the material trying to avoid waste.
James
James
I love all the math/geometry challenges that come up on here. LOL
Turns out, while I love the rigorous trigonometry, the relationships actually resolve to a very simple formula. For a circle INSIDE the octagon (in eyekode's example):
SIDE LENGTH = DIAMETER / (1 + sq rt of 2) or approx.:
SIDE LENGTH = DIAMETER / 2.414
giving approx SIDE LENGTH = 8/2.414 = 3.314
Got this by solving the formulas on the bottom of the page Joe referenced. For anyone interested in using that calculator, it seemed very devoid of any instructions.... The two variables of interest are "Excircle radius" or the "Incircle radius" (depending on if the circle is outside or inside the octagon, respectively). Using eyekode's example (and most excellent drawing:thumbs_up), you would enter "4" in the "Incircle radius" field (4' being the radius of the 8' circle). That gives "Edge length" = 3.314.
BTW, the formula for a circle OUTSIDE the octagon is approx:
SIDE LENGTH = DIAMETER / 2.613
Turns out, while I love the rigorous trigonometry, the relationships actually resolve to a very simple formula. For a circle INSIDE the octagon (in eyekode's example):
SIDE LENGTH = DIAMETER / (1 + sq rt of 2) or approx.:
SIDE LENGTH = DIAMETER / 2.414
giving approx SIDE LENGTH = 8/2.414 = 3.314
Got this by solving the formulas on the bottom of the page Joe referenced. For anyone interested in using that calculator, it seemed very devoid of any instructions.... The two variables of interest are "Excircle radius" or the "Incircle radius" (depending on if the circle is outside or inside the octagon, respectively). Using eyekode's example (and most excellent drawing:thumbs_up), you would enter "4" in the "Incircle radius" field (4' being the radius of the 8' circle). That gives "Edge length" = 3.314.
BTW, the formula for a circle OUTSIDE the octagon is approx:
SIDE LENGTH = DIAMETER / 2.613
For those of us whom have become "memory challenged" from time to time with regard to shop math, there are two shop references I would recommend which conveniently summarize the math necessary to design, as well as instructions on how to mark and layout, common geometric shapes (as well as other useful shop math and miscellany). Both are published by Popular Woodworking.
Book #1: "Practical Shop Math" by Tom Begnal, ISBN 978-1-55870-783-2, Published 2006, $14.99USD. * (Your question answered on page 170)
Book #2: "Pocket Shop Reference" by Tom Begnal, ISBN 978-1-55870-782-5, Published 2006, $14.99USD. * (Your question answered on page 47)
Another book with some useful charts, though not nearly as much so as the two above, is "Woodworker's Pocket Reference" by Charlie Self, ISBN 1-56523-239-9, Published 2005, $14.95USD.
* ISBN numbers are currently in transition from a 10-digit system to a 13-digit system. The number provided is the newer 13-digit ISBN number. If you require an equivalent older 10-digit ISBN number please subtract the first three digits (978) from the above.
I have no connections with any of these book's authors or publishers, but I have found them to be useful references from time to time and feel others might also find them to be useful. Especially those of us who don't necessarily deal with complex shapes on a day-to-day basis or have gotten a little rusty with our trigonometry.
Book #1: "Practical Shop Math" by Tom Begnal, ISBN 978-1-55870-783-2, Published 2006, $14.99USD. * (Your question answered on page 170)
Book #2: "Pocket Shop Reference" by Tom Begnal, ISBN 978-1-55870-782-5, Published 2006, $14.99USD. * (Your question answered on page 47)
Another book with some useful charts, though not nearly as much so as the two above, is "Woodworker's Pocket Reference" by Charlie Self, ISBN 1-56523-239-9, Published 2005, $14.95USD.
* ISBN numbers are currently in transition from a 10-digit system to a 13-digit system. The number provided is the newer 13-digit ISBN number. If you require an equivalent older 10-digit ISBN number please subtract the first three digits (978) from the above.
I have no connections with any of these book's authors or publishers, but I have found them to be useful references from time to time and feel others might also find them to be useful. Especially those of us who don't necessarily deal with complex shapes on a day-to-day basis or have gotten a little rusty with our trigonometry.
- Status
