Find exact width of cutting a board into N equal pieces January 10, 2012
Posted by jisommer in : Email Questions , comments closedQuestion:
A friend asked me to help him cut a 10″ board into 10 equal parts factoring in for the loss due to the 1/8″ blade kerf. We scratched our heads and wished we would have paid more attention in math class.
We knew that we would have to make 9 cuts to get the 10 pieces. So 9 X 1/8″ = 1 1/8″.
So to approximate the cutting width we subtracted 1/8″ from each piece and cut the pieces 7/8″ inches wide. This was not exact but fairly close.
My questions is what math formula should I use to get the exact width of cutting a board into equal pieces factoring in the loss of the saw kerf? Thanks
Solution:
You logic will as you said get you pretty close to the correct cutting width.
Mathematically to calculate the cut width:
10*x + 9(1/8) = 10
10*x = 10 – 9/8 = 80/8 – 9/8 = 71/8
x = 71/80 which is very close to 7/8.
Many times in the wood shop a ruler is not the most accurate means of measurement unless our measurements are to the nearest 8th or 16th.
In order to get a more accurate measurement of 71/80 see this diagram for details.
It shows how to mark on a piece of paper a length of 71/80 using standard ruler and the concept of similar triangles from geometry.
See this page on my site for a brief explanation.
This technique is used many times to divide a line into a given number of equal segments.
In this case by multiplying by 10 and dividing into 10 equal pieces we get an accurate length of a decimal quantity.
Gear Math June 30, 2010
Posted by jisommer in : Uncategorized , comments closedHere is a link to an article from Make Online Magazine with information about mathematics of Gears.
http://blog.makezine.com/archive/2010/06/make_your_own_gears.html
Compass Rose October 23, 2009
Posted by jisommer in : Email Questions , comments closedQuestion:
John… I want to make a 36″ dia. compass rose with true north 8 points. I can’t find a plan
that gives me the angles. Do you know of any information that would be helpfull. I have been
doing woodworking for a while, I am 74 and keep active in my shop but decided to make
small projects and not furniture like I did. thanks Dan Lober
Solution:
Here is an image of an 8 pt compass rose. The angles are independent of the radius of the circle.
8pt Compass Rose
Also for other number of points:
| N | NbyE | NNE | NEbyN | NE | NEbyE | ENE | EbyN |
| 0 | 11.25 | 22.5 | 33.75 | 45 | 56.25 | 67.5 | 78.75 |
| E | EbyS | ESE | SEbyE | SE | SEbyE | SSE | SbyE |
| 90 | 101.25 | 112.5 | 123.75 | 135 | 146.25 | 157.5 | 168.75 |
| S | SbyW | SSW | SWbyS | SW | SWbyW | WSW | WbyS |
| 180 | 191.25 | 202.5 | 213.75 | 225 | 236.25 | 247.5 | 258.75 |
| W | WbyN | WNW | NWbyW | NW | NWbyN | NNW | NbyW |
| 270 | 281.25 | 292.5 | 303.75 | 315 | 326.25 | 337.5 | 348.75 |
Here is a link to some instructions on drawing a compass rose:
Rise and fall – degrees calculation
Posted by jisommer in : Email Questions , comments closedQuestion:
I don’t know if I’m saying this correctly and using the correct terms. I can’t find the answer, and am having trouble figuring it out for myself. It seems like there should be a very simple formula to calculate this.
I am building a ‘veranda’ for my tortoise, that will go over his dog house, from a 4′ x 8′ sheet of plywood and 2×4’s. I want it to be at an angle so that the rain will run off. So I thought I would make it lower one inch for every 16 inches. Eight feet is 96 inches, and 16 goes into 96 six times, so one side will be 4′ tall and the other 3′6″ tall.
What I need is a formula that will take “rise and fall” (?) of X and Y (x=16, y=1) and convert it to the degrees that I can set my table saw to so that I can cut the correct angle for the 2×4’s.
The resulting overall length won’t actually be 8′ because of the angling, so I don’t know if that means anything, or needs to be taken into account, or what.
Attached and inserted is a diagram image.
Question Illustration
Thanks.
Bill
Solution:
To find the angle you need to use some trigonometry.
In your diagram we have a right triangle with a hypotenuse of 96″ (8 ft) and an adjacent leg of 4″ (difference between 4′ and 3′ 6″). Since you know the rise and fall is a ratio of 1 to 16 you can find the angle by computing the arc cosine of 1/16 using a calculator [this ratio is what determines the angle, so if the length is not 8ft but you still want this slope for the over hang then still use the 1 to 16 ratio].
For example, if you go to this site http://www.carbidedepot.com/formulas-trigright.asp you will find a basic right triangle calculator. See attached picture for illustration. Enter 16 for side c and 1 for side a (or could use 96 and 4) then click calculate and the site will return the angle you requested which in this case is 86.42 degrees.
If you cut a smaller board in the same ratio as your bigger project, such as 1″ tall and 16″ long in the form of the right triangle then you will have a template for the angle you need without doing any mathematics. This is because of the mathematic’s property of similar triangles which is the basis of trigonometry – the ratio of sides and angles are the same as you make a triangle bigger and smaller using proportional increase or reduction in size.
This link to Wolfram-Alpha will give you information on equations of the right triangle.
Curved chest top
Posted by jisommer in : Email Questions, Uncategorized , comments closedThis question builds on the previous post which shows how to calculate the center of a circle given a chord and distance from the chord to the circle (called the Saggita).
Question:
Hi,
i’m working on a wood chest and the lid i wanted it arched but i have no idea at what angle and HOW MANY pieces of wood i need to complete the lid.
My chest lid BASE measure 27 inches and the height at the middle point it should be 7″…. the stripes are about 1/2″ thick and 3″ wide. HOW do i do it?
PS: a small draft is attached to the message!
Thank you so much!
William!
Solution:
Here is a link to a GeoGebra simulation I created to illustrate how to answer the question.
Below is a picture from William with some additional lines I added to help to illustrate how the mathematical solution does apply to his original question. Note – the height stated as 17″ is incorrect, should be 7″. 17″ is more than half the length of the chord.
Here is a snapshot of a Google Spreadsheet I created to work out the calculations with his specific information. Note D1 reference is for value of PI.
Chest Top Calculations
I also found a nice simple calculator for circle, arcs and chords which can also be helpful.
Calculate Radius of Arc March 26, 2008
Posted by jisommer in : Reference, Uncategorized , comments closedMany times you need to calculate the radius of a circular arc for a given chord width and height (distance from chord to the top of the arc) – See diagram below:
On the diagram w represents the width of the circular arc (chord) and d is the height (distance from chord to top of the arc). Given both of these we can calculate the radius of the circle needed to create the arc from using the formula on the diagram designated as r.
Here is a calculator to compute the radius:
The formula is derived using Analytical Geometry. The chord and arc are drawn on the XY axis with one end of the arc at point (0,0) and the other end at (w,0). The center of the arc will be at point (w/2, d). Using the standard equation of the circle and algebra we can derive the formula for the radius needed to create the arc. Below shows the derivation of the formula. Click on image to view larger.
Compound Miters August 20, 2007
Posted by jisommer in : Reference, Uncategorized , comments closedCompound miter cuts involve making a miter angle cut and a blade tilt cut for projects such as crown moulding, hexagon shaped container with sides tilted out, etc. Some of these cuts require not only a standard miter angle (anlge 1 in illustration) in a vertical direction but also a horizontal angled cut called a bevel (angle 2). 
I have found several very good sites which are good resources with tools, explanations and mathematics for compound miter cuts and application.
Â
Â
Â
Â
Â
Â
Â
This site has a Advanced Box Cutting calculator to determine the proper cutting angles for polygons. And here is the page for the mathematics of Dihedral and Equal Angle Polygons (regular).
This is a link to a PDF with another explanation of the mathematics of dihedral angles for the woodworker.
A site with a calculator specific for a pyramid.
The same author also has good explanations, tables and a calcualtor for cutting crown molding.
For additional information and instructions on Crown Molding see Alter Eagle site or Dewalt site.
This is a link to a Dr. Math article about Pyramid Construction in which he includes some of the mathematics and a good way to visual a compound angle and its mathematics.
Taper Jig May 31, 2007
Posted by jisommer in : Email Questions, Uncategorized , comments closedOriginal Question:
I am constructing a taper jig to use with my ShopSmith woodworking machine. Is there an already derived formula for conversion of taper (inches per foot) directly to degrees and vise versa. I used the tangent of the opposite side (taper) divided by one foot (12 inches, adjacent side of triangle) and then look up the degrees for the derived tangent for my right triangle. This is the answer but I thought perhaps you already have a quick and easy formula for this. Your help is appreciated. I really like your website. It has much useful information.
Answer:
This is an illustration of a typical taper jig used on a table saw. The pictures below are pages describing the mathematics of a taper jig and suggestion how to add a measuring device to set the jig taper directly in inches per foot.
Reader’s response.
Thought you might like to see my completed Taper Jig (picture1, picture2). I used a 24 inch long board next to the work instead of a 12 inche one, so my separation of the “V” is in 1/2 inch increments instead of quarters of an inch.
i.e. 1/4 inch per foot on 12 inches is same as 1/2 per foot on 24 inch length.
I used tangent to calculate the angles in degrees for each setting. My scale goes from 1/4 inch per foot to 2 1/2 inches per foot and from 1.16 to 11.57 degrees. Note the taper in inches per foot is the left column and the degrees are in the right column.
Again, thank you so much for your help. I think I was on the right track, just need a little reassurance and coaching.
Curved Deck Rail
Posted by jisommer in : Email Questions, Uncategorized , comments closedOriginal Question:
My deck has a curve in it. It’s like a half circle, except its depth is not equal to its width. From a bird’s eye view, the curve is 72″ wide and 48″ deep. The deck posts are already set.
There are posts at the point where the curve begins and ends and there are two additional posts set at equal intervals, dissecting the curve into three parts.
To cut the top and bottom 3.5″ wide rails (2×4 lumber on the straight rail) I will use 2X6 lumber and draw a pattern to produce the three sections of curved railing.
To cut the rail cap (5.5″ wide, or a 2X6 on the straight rail cap) I will use a 2X8 or 2X10 to draw a pattern to produce the three sections of curved 5.5″ wide rail cap.
I can create the curve for the railing in two ways:
- Transfer the curve of the facia board (that covers the deck framing’s joist ends) to the deck rails?
- Calculate the railing curve, as dictated by the post placement and curve dimensions.
Here is my question:
How do I do the easiest of the two methods?
Or, is there an easier way to calculate the rail curves so I can draw my patterns?
Answer:
Diagram of the representation of the curved deck railing:
Calculations:
GIVEN:
Width of curve of deck 72″
Depth of curve 48″
Width of rail 5.5″
LEGEND:
R = radius of deck curve. [37.5"]
a = angle of arc of 3 equal sections of deck rail. [70.8o]
V = added length for rail width
Z = length of cord of rail. [43.45" + 2* 9.04"]
W = minimum width of board needed to cut curve. [6.93" + width of the rail 5.5" ]
V = additional width to W (to draw pattern for curve) [6.75"]
Bird Feeder
Posted by jisommer in : Email Questions, Uncategorized , comments closedOriginal Question:
I’m into building bird feeders(see picture ) and need to know if there is a formula to calculate the length and the width of the pie shaped pieces that make up the roof, I would like to keep the roof about 16″ wide and about 5″ high using 10 pie or (pyramid) shaped pieces per roof.
If you have a better or easier way to compute the sizes I’m all ears.
Answer:
Diagram 1 is a “birds eye” view of the bird house. The house has ten sides with radius of 8″.
Angle AOB will be 36 degrees (360/10). We can calculate AB as follows:
Bisect AOB with a perpendicuar to AB. This will also bisect AB.
sin 18 = BC / 8 = AC / 8
BC = 8 * sin 18 = 2.4721
AB = 2 * BC = 4.9443″ which will be the length of the base of each triangle forming the roof.
Diagram 2 is the side view of the bird feeder.
OT is the inside height which was given to be 5″.
The base is 8″ (half the diameter given – 16″)
AT2 = OA2 + OT2 = 64 + 25 = 89
AT = 9.434″
Tan (Angle AOT) = 5 / 8 thus
Angle AOT = 32 degrees
Reader’s Response:
Thank You so much John, When I saw your trig formula for figuring this out for me a light when on and my math days started coming back, this will help me enormously, Thank You again. –Kenneth