5. Golden Ratio
6. Fractions & Equivalent Decimal Values
- Pi = 3.14159 = Circumference / Diameter
- Area = Pi x Radius x Radius = 3.14159 x Radius x Radius
- Circumference = 2 x Pi x Radius = 6.28 x Radius
- 360 degrees in a Circle.
2 — Triangles
3 — Hexagons
- Perimeter of Hex = 6 x Radius of Circle (That Hex is inscribed inside of)
- Each side of Hex equals radius of Circle
- For the Sketch below, the line “AB” is drawn such that it runs thru the center of the circle shown. As an example, assume the radius of the circle is 2″. Then the Hexagon perimeter will be 12″ (12″ = 6 x 2″), and the length of each side of the Hex will be 2″.
Notes for the Sketch above (“Drawing a Hexagon”)
Note 1 — Swing an arc from pivot point “A” with a radius equal to the radius of the circle shown.
Note 2 — Swing an arc from pivot point “B” with a radius equal to the radius of the circle shown.
Note 3 — Draw a line between point “A” and where the arc intersects the circle.
Note 4 — Draw a line between point “B” and where the arc intersects the circle.
Note 5 — Draw a line between the points where the arcs intersect the circle.
Additional Hexagon data is included in the following Sketch.
4 — Octagons
Unfortunately for Octagons, the Octagon perimeter and length of each Octagon side do not relate well to the parameters of the circle (In the Sketch below). By this, I mean that the relationship between the circle radius and the Octagon side lengths is not a nice whole number type of relationship. But no problem, we will just use a slightly different approach, to construct an Octagon.
General Notes for “Drawing an Octagon”
- Starting with the circle shown, draw lines “AB” and “CD” as shown. The two lines must pass thru the center of the circle, and the two lines must be at a right angle to one another.
- Bisect the two lines just drawn with lines “EF” and “GH”, as shown in the Sketch above.
- Now draw the Octagon by drawing the lines “AE”, “EC”, “CG”, and so on around the circle, as shown in the Sketch above.
Note the 45 degree angles, angles = 45 degrees each (8 x 45 degrees = 360 degrees).
It’s nice to have the Octagon perimeter length, for knowing how long a board you will need to frame an Octagon shaped perimeter. Now the perimeter length issue can be handled in one of two ways.
- If the Octagon has been drawn, then measure one of the Octagon sides and multiple the length by 8 (Perimeter = 8 x one Octagon side length).
- If the Octagon has not been drawn, but you know the radius of the circle that the Octagon will be inscribed inside of. Then compute the circumference of the circle; the circle circumference will be slightly greater than the Octagon perimeter length.
5 — Golden Ratio
- For dimensioning furniture, the Golden Ratio a constant which equals approx. 1.618 is commonly used for proportioning the height, width, & depth.
- As an example, assume you want a width of 20″ for a wall hung cabinet:
1) Then height = 1.618 x 20″ = 32.36″
2) And depth = 0.618 x 20″ = 12.36″
- Personally, I would use 1.6 and 0.6. Simplifies the calculations. Therefore for the above cabinet, the height would be 32″ (1.6 x 20″) and the depth would be 12″ (0.6 x 20″). And the resulting cabinet would be 32″ (Height) x 20″ (Width) x 12″ (Deep). Since the front is the predominantly viewed face, decreasing or increasing the depth should not be a major ratio pleasing issue.
- 3/2 (or 1.5), 5/3 (or 1.67), 8/5 (or 1.6), & 34/21 (or 1.619) are other ratios that have been used in the past, to obtain pleasing ratios.
- Golden Ratio is the ratio you want to see for the dominate face of your furniture that the viewer will see. Most likely width by height. Odds are getting all three (maybe even just two dimensions) of your dimensions (width x length x depth) to be perfectly proportioned is not going to happen. But then, function should trump a perfect Golden Ratio.
- In the end, the Golden Ratio seems to be more of a ratio between our dimensions that we strive for. And again, function trumps a perfect ratio between dimensions.
Dimensions for length, width, and depth, based on an approximation of the Golden Ratio (using an 8/5 ratio) are listed in the following Table.
Notes Regarding the Table above:
- Smallest of 3 dimensions = 0.6 x (Medium of three dimensions)
- Largest of 3 dimensions = 1.6 x (Medium of 3 dimensions)
6 — Fractions & Equivalent Decimal Values
Converting in Your head — 3/8, 5/8, & 7/8 Values to Decimal Format
If you remember that 1/8 = 0.125, then it’s fairly easy to add 1/8 to 1/4, 1/2, and 3/4, in your head to compute the decimal values for 3/8, 5/8, & 7/8. For example:
3/8″ = 1/4″ + 1/8″ = 0.250″ + 0.125″ = 0.375″
5/8″ = 1/2″ + 1/8″ = 0.500″ + 0.125″ = 0.625″
7/8″ = 3/4″ + 1/8″ = 0.750″ + 0.125″ = 0.875″
In closing, I hope you found the above eclectic collection of math data to be useful. I know I find the above math data handy.
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