O.K., it's been a while since some of us have had our formal math training, and those skills can get rusty. Here is your opportunity to revisit a few basic concepts to help with your common layout and design tasks. I'm not here to give an exhaustive treatment in mathematics, and I do not want to throw a lot of unnecessary equations or calculations at anyone. This web page isn't intended to teach a math course. Its purpose is to present the quick-and-dirty math rules and formulas to give woodworkers practical help with some common design problems. I've included a few cheesy, hand-drawn diagrams for illustration. Enjoy.
Using a right triangle to determine a taper, miter, or bevel.
When cutting or milling a taper, bevel, miter or chamfer on a piece of stock with perpendicular surfaces, a right triangle can be used to determine the angle. For example, when cutting a taper in a straight table leg, the cut off piece can be represented by a right triangle. You can apply this when cutting a bevel or miter, or a dog-ear off a corner.
Consider the triangle shown in Figure 1 above. The three corners of this triangle are represented by capital letters A, B, and C. Based on the angle at A we have named the three sides: adjacent (a), opposite (o), and hypotenuse (h). When exactly one of the three corners (B) is perpendicular (90º), it is considered a right triangle. The other two corners (A and C) are acute angles - each being less than 90º. The sum of the three angles will always be 180º. Thus, for example, the three corners of a right triangle with a 45º hypotenuse will have 45º-45º-90º angles, the sum of which equals 180º. Since one corner (B) is always 90º, it follows that the sum of the two acute angles (A and C) will always equal 90º.
In Figure 1, the angle at point B is 90º (right angle). There is a geometric relationship between the three sides of the triangle. The square of the hypotenuse (h) is equal to the sum of the squares of the adjacent (a) and opposite (o) sides. This is known as the Pythagorean Theorem. Using the Pythagorean Theorem, we can solve for any unknown side of a right triangle if we know the lengths of the other two.
Solving for the hypotenuse:
Thus, for example, if we know that the adjacent side is 4 inches in length and the opposite is 3 inches, the hypotenuse is solved:
This is the familiar "3-4-5" right triangle.
If we know the length of the hypotenuse and the opposite, we can solve for the adjacent side:
Similarly, if we know the hypotenuse and adjacent, we can solve for the opposite:
If the right triangle has a hypotenuse sloping at 45º, then the adjacent and opposite sides are equal. Solving this triangle is easier, and does not require using the Pythagorean Theorem at all. There is merely a ratio between the length of the hypotenuse and the other sides:
Conversely, the adjacent and opposite sides are determined:
Since the hypotenuse of a 45º right triangle is the same as a diagonal line bisecting a square, the above equations can be used to determine the diagonal of a square.
Now for a little basic trigonometry.
We saw from applying the Pythagorean Theorem above that we can determine the length of a third side of a right triangle when we know the lengths of the other two. But sometimes, it is necessary to know the acute angles when all we have are the dimensions of the sides, such as when we want to set the angle on a miter saw or taper jig to produce a certain cut. Don't be daunted by this -- all you need is a calculator with trig functions: sine (sin), cosine (cos), tangent (tan); and the inverse trig functions: arcsine (asin), arccosine (acos), and arctangent (atan). Here are the trigonometric identities for a right triangle:
Let's take the three equations above, and solve for angle A:
As you can see, you can solve the angle of corner A if you know the lengths of any two sides. For example, if we know that the opposite side is 3 inches in length and the adjacent side is 4 inches, we can determine the angle A using the third trig identity from above:
Since the sum of the two acute angles equals 90º, we can now determine the angle of C by subtraction:
When you have a right triangle, and you know one of the acute angles and the length of only one side of the triangle, you can solve the lengths of the remaining two sides and the other acute angle (C).
Solving for the adjacent side:
Solving for the opposite side:
Solving for the hypotenuse:
Note: Keep in mind that the above equations apply only to right triangles. If a triangle is any other type (no corner at 90º), those equations will not work.
Creating an octagon.
Figure 2 above shows an octagonal shape superimposed onto a square. This eight-sided polygon is what is called a regular octagon: all 8 sides are equal in length and all 8 corners are at equal angles. In fact, it is easy to visualize this octagon as simply a square with the corners lopped off at 45º angles. The tricky part is determining how much to cut off. Of course, you don't have to make your octagon regular; you can trim the corners to create one of your liking - an irregular octagon.
When laying out a regular octagon on a square piece of stock, we need only decide how much to cut off at the corners. I'll spare you the pain of the trigonometry involved and just express the dimensions in ratios.
From Figure 2 above, the square used in this design has its length and width denoted by "S". Note that each corner to be removed is really a small right triangle with a hypotenuse sloped at 45º. Thus we can use "h" to represent each side. Accordingly, the other two sides of each of the triangles (denoted by "a") can be used as a measurement of how far inward from the square's corners to mark the 45º cuts. This amount "a" is determined to be:
Thus, for example, if we are to design an octagonal table top that is 36 inches in width (distance between two opposing sides, not corners), we start with a square piece of material, 36 by 36 inches. Apply the ratio to determine "a":
We then make marks at 10.55 inches inward from the corners of the square, and draw lines to mark the locations for the cuts. The resulting sides will be uniform, and of a length of 0.414·S. The length "h" of the sides of a 36" wide octagon will be:
Additional note: If cutting an octagon from a round piece of stock, simply measure the circumference with a measuring tape or piece of string. Divide this circumference by 8, and make marks at these 1/8 intervals. Connect the marks and cut. However your octagon will be smaller in width than the diameter of the original circle. Divide the circle diameter by 1.082. Thus, a 36" diameter circle will yield an octagon 33.27" in width.
Creating a hexagon.
Figure 3 shows a hexagon superimposed onto a rectangle. The hexagon is "regular," in that all the sides are equal in length and all corners are of the same angle. Unlike the octagon discussed above, the layout of a hexagon will require a sheet of material that is a bit greater in length than width, due to the position of two of the corners. In fact, the ratio of length to width is:
Thus, for example, to layout a hexagonal tabletop that is 36 inches in width (across two opposing sides, not corners), you will need a piece of stock 36" wide and 41.58" long.
Looking at Figure 3, you can see that the pieces to be removed from the rectangular workpiece comprise 30º-60º-90º triangles. It is easy to determine where to mark for the cutoffs. Again, I've spared the trig and will provide dimensions in ratios.
Marking the left and right corners is simply a matter of measuring halfway down the sides along the vertical edges. Dimension "a" can be determined as a ratio of the length "L" (horizontal) dimension as follows:
This makes the layout of a hexagon easy to remember: side corners at the halfway point along the width, top corners 1/4 way inward along the length. Draw lines to connect the marks and you are ready to cut.
Additional note: If cutting a hexagon from a round piece of stock, simply measure the circumference with a measuring tape or piece of string. Divide this circumference by 6, and make marks at these 1/6 intervals. Connect the marks and cut. However your hexagon will be smaller in width than the diameter of the original circle. Divide the circle diameter by 1.155. Thus, a 36" diameter circle will yield a hexagon 31.17" in width.
Some common geometric formulae.
Circumference of a circle:
Area of a circle:
Volume of a cylinder:
Surface area of the sides of a cylinder:
Diagonal of a square:
Diagonal of a rectangle:
Diagonal of a box: