## Geometry- a tool for the carpenter

**Geometry for the woodworker and the carpenter.**

I use math nearly every day in my finish carpentry business.

In woodworking and cabinetry it's essential.

Along with all my other tools, I carry "cheat sheets" with various formulas with me. they are as important as any.

**Any kind of carpentry work involves geometry**

– it’s all about angles and levels and circles with their radii and diameters.

It's a HUGE advantage to be able to figure a diagonal measurement when doing layout work for a foundation, and it usually impresses other tradesmen if you have an accurate answer ready for a problem they can't solve by trial and error.

Geometry is extremely handy and I thought others would like to refresh some basic geometry skills also.

The images are from a Dover publication called the *The cabinet-makers assistant* originally published in 1853.

**Problem 1 - At a given point on a straight line draw a line perpendicular to it. **

Draw a line AB. Point C is equal distance from both ends.
From C scribe an arc whose radius is greater than CA. Using the same radius scribe an arc from B in the same manner. Draw a straight line from c through the point of the intersection of the arcs.

**Problem 2 - From a given point let fall a straight line perpendicular upon a given line. **

Let C be the given point and AB the given straight line.With C as center scribe an arc of any radius that will cut line AB at D and E. From D and E as centers and with a radius greater than half of DE scribe two arcs that intersect at F. Draw the perpendicular line DF.

**Problem 3 - From the extremity of a straight line draw a perpendicular line to it. **

Let the straight line be AB and the extremity point be B.
From any point C scribe an arc that is greater than a semicircle that will intersect the line at D and B. Through D and C draw a straight line to intersect the arc at E. Join E and B with a straight line.

**Problem 4- Bisect(divide into 2 equal parts)a given angle.**

The given angle is ABC. From vertec B, with any radius BD scribe an arc DE to intersect the outsides of the given angle. From D and E as centers and with a radius greater than half of DE scribe two arcs to intersect at F. A straight line drawn from B through F will bisect the given angle.

This problem may also be solved by rectilineal contruction using a ruler;

BAC is the given angle. On side AB make two points D and E. On side AC make AF equal to AD and Make AG equal to AE. Joine DG and EF with straight lines. They intersect at H. A straight line drawn from A through H will bisect the given angle.

**Problem 5 - Divide a right angle into 3 equal parts.**

ABC is to be the right angle. From vertex B scribe an arc with any radius to intersect the right angle sides at D and G forming arc DG. Using the same radius scribe an arc from D intersecting DG at F and from G an arc intersecting at E. Straight lines from B throught E anf F will trisect the angle ABC.

**Problem 6 -Construct an equilateral tringle on a given straight line. **

AB is the given base line. From the ends of the line and using points A and B as centers scribe an arc using AB as the radius scribe arcs intersecting at C. Straight lines from A to A and B to C will form an equilateral triangle.

**Problem 7 - Construct an isosceles triangle from a given base line. **

AB is the given base line and MN is to be equal to the other two sides of the triangle. Using MN as the radius scribe arcs from points A and B to intersect at C. Draw lines from A and B to C.

**Problem 8 - Divide a board into equal rips. **

Let the board be ABCD and the parallel sides be AB and CD,and whose width is 9".

DIvide it into 7 rips. Lay a rule diagonally acros the board.
7 inches will not extend across the board but if half of 7, or 3 1/2 is as added, then 10 1/2 wiil extend diagonally. This dimension can be dived into 1 1/2" along EF and then along HG.

Lines drawn through these marks will be equidistand and parallel.

**Problem 9 - Through a given point draw a line parallel to a given line AB. **

From point C, the given point at a required distance fron AB, draw a line meeting AB at a point D.

From D, with the radius DC, scribe an arc meeting AB at E.

From C as the center with the same radius Scribe an arc DF.

At d scribe an arc to F with the radius of CE,

Through C anf F draw a straight line that is parallel to AB.

**Problem 10 - Draw a line parallel to AB with a given point C at the required distance to AB by rectilineal construction. **

From C draw a straight line to intersect AB at any point D and extend it until DE equals DC.

From E draw a straight line to intersect AB to F and extend this line to G until FG equals EF.

Through C and G draw a line that will be parallel to AB.

**Problem 11 - Divide a line into any number of equal parts. **

BC is the given line and divide it into 5 equal parts.

From the extremity of B, draw a line BA, forming any acute angle with BC. On BA measure 5 equal parts from B to A. Connect AC with a straight line.

From each of the other points on BA draw parallel lines to BC. This will divide BC into equal parts.

**Problem 12 - Creating a regular hexagon with a ruler and compass**

**Problem 13 - Construction of a regular pentagon with a ruler and compass **

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