When the Three Lengths Are Known (3-4-5)

1. Draw a line equal to one of the known lengths.
2. Set your compass to the second known length (6 inches) and swing an arc from one end of the line (point A).
3. Set your compass to the third known length (3 inches) and swing an arc from the other end (point B) of the line, crossing the first arc.
4. Complete the triangle by drawing two lines from the ends (points A and B) of the line to the cross point of the two arcs.

1. Draw a line
2. Set your compass to approximately ¾ of the length of the line. (The important thing is that it is set to more than half. The further away from half way, the more accurate, but the larger the space required becomes.)
3. Keeping this distance set on your compass, swing an arc from both ends of the line towards the centre of the line.
4. Draw a line, connecting the two intersection points of the arcs.

1. Draw an arc.
   
2. Draw a line (chord) that connects the two points of the arc (now, bisect the line).
   
3. Set your compass to approximately ¾ of the length of the line. (The important thing is to set it to more than half. The further past half way, the more accurate, but the larger the space required becomes.)
4. Keeping this distance set on your compass, swing arcs from both ends of the line, directed towards the centre.
5. Draw a line through the two intersection points of the arcs.
   

1. Set your compass to a radius of reasonable distance inside the angle and swing it from the vertex to create an arc inside the angle.
2. . Draw a line (chord) from the two points that cross the lines that make the angle.
3. Now, bisect the line (chord). You can use the vertex as one point, so you only need arc outside the chord. Set your compass to approximately ¾ of the length of the line.
4. Keeping this distance set on your compass, swing arcs from both ends of the line.
5. Draw a line through the vertex to the intersection of the arcs.
   

1. Draw the angle to be copied and label it A-B-C. Then, draw a baseline for the new angle and label it A-B.
2. Set your compass to a radius of reasonable distance. Swing it from the vertex point of the original angle and from point A of the new baseline.
3. Pick up the distance of this arc from the original angle and transfer it to the copied angle.
4. Draw in the line from point A to the intersection point to form the angle. Pick up distance A-C and transfer if required.

1. Draw a line.
   
2. Set your compass equal to the distance required for the parallel line.
3. Swing at least two arcs from the line.
   
4. Create the parallel line from the peak (point of tangency) of the arcs.

Create a Perpendicular Line at the Middle of a Line

1. Draw a line.
   
2. Bisect the line.
   
3. In bisecting the line, we have created a perpendicular line (x-y axis).

Create a Perpendicular Line at the End of a Line

1. Draw a line.
   
2. Set your compass to a radius of reasonable distance.
   
3. Swing that arc from the end of the line.
4. Using the same distance (radius) on your compass, swing that distance along the arc twice. Each time you swing the radius along its own arc/circumference, it create a 60° angle and sector.
5. Bisect the arc between the last two points. We are bisecting the second 60° sector to create two 30° sectors. If we then look at the arc, we have a 60° sector and a 30° sector, which combine to make a 90° quadrant.
6. From the end of the line, draw a line through the last point.

1. Establish three random points. (Under normal conditions, these points will not be random).
2. Draw two lines to connect the points.
   
3. Bisect the two lines and create perpendicular lines.
4. Where the two perpendicular lines cross becomes the radius point for the arc.
5. Set your compass from the established radius point to one of the three points.
6. Swing the arc through the points.
   

1. Draw an x-y axis.
   
2. Draw a circle.
   
3. From each quadrant point, swing the radius in both directions to cross the circumference. (Remember, when we swing the radius along an arc of equal distance, we create 60°.) As we do it both ways, we end up with 30°-60°-90° all the way around the circumference.
   

1. Draw a rectangle.
   
2. Determine a larger length that divides easily into how many equal spaces you want. (For example, if four equal spaces are required, 10 inches, 12 inches, 16 inches, etc. are good choices.)
3. Angle the ruler so that zero and the number chosen, in this case 10 inches, are on the vertical, outside edges of the rectangle (you may need to extend the vertical lines of the rectangle).
4. Mark the divisions along the ruler and square them vertically. If four divisions are required and 10 inches is chosen, then mark every 2.5 inches on the ruler.
   

If we think of a hexagon inside a circle, it has a radius equal to the length of any given side. This makes it quite easy to construct.

1. Set your compass to the desired radius or side length.
2. Draw a circle
   
3. Using the same radius, swing it along the circumference six times. Remember, this creates 60° sectors.
4. Connect the points to create the hexagon.
   

1. Choose how many side you want.
2. Choose a side length. Think of this line as a radius.
3. Draw a baseline double that length.
4. Set you compass to the side length (radius).
5. Draw a half circle from the centre of the line.
6. Divide the half circle into the number of sides required and number it (other geometric construction techniques will be used here).
7. Draw a line from the radius point to the third point and label the angle/triangle A-B-C. This will create an angle equal to the polygon angle (for example, five sides equals 5 ÷ 360 = 72°).
8. Use “Draw an Arc Through Three Points” to find the radius point for A-B-C.
9. Use this radius to draw a complete circle.
10. Set your compass to the required side length and swing it along the circumference however many times needed to complete the polygon.

1. Draw a full plan (top) and elevation view (front) of the project, complete with the miter line.
2. Divide the plan view into 12 equal parts (see Divide a Circle Into 12 Equal Parts) and label it. It is common practice to label a round object with numbers.
3. Project the plan view divisions down into the elevation view.
4. Draw the stretch-out directly to the right of the elevation view and divide it into 12 equal parts (see Divide a Line Using a Ruler on an Angle). Label it to match the plan view and make sure to start the labeling where you want the seam. Add any required seam allowances outside of the stretch-out.
5. At the points where the element lines cross the miter line, project them into the stretch-out.
6. Following the labeling, circle the intersection points on the stretch-out. Don’t put a dot over the points, but circle around them.
7. Using a flexible curve, join the points to draw in the miter line, completing the pattern.

1. Draw a full front elevation view and end elevation view (right side). Notice that the tee does not go passed centre of the pipe, it never will. Because of that, we can delete that portion of the drawing.
2. Draw profiles on the tee in both views and divide into 6 equal parts (see Divide a Circle Into 12 Equal Parts). Include all labeling. Keep in mind that the labeling will rotate 90° with the view.
3. Draw in the element lines from the profile divisions.
4. Where the element lines hit the pipe in the end elevation view, project them horizontally into the front view.
5. Following the labeling, draw in the miter line in the front view.
6. Draw the tee stretch-out and divide into 12 equal parts (see Divide a Line Using a Ruler on an Angle). Label the stretch-out to match the elevation views.
7. Transfer/project the corresponding points from the elevation views to the stretch-out.
8. Use a flexible curve to draw in the miter line, completing the pattern.

1. Calculate the blank size. Use the stretch-out for one dimension (horizontal) and the seam height plus the radius of the pipe for the second (vertical) dimension. Add any seam allowances on before shearing the blank size.
   
   
2. With your dividers, swing the radius of the pipe to 180° at the bottom of the blank. Then, from the same radius point, swing the radius of the tee and divide it into 6 equal parts (see Divide a Circle Into 12 Equal Parts) and label it. Remember that only the tee is divided. Also, notice that the half circles are symmetrical, so a quarter circle is the minimum required in this case.
   
   
3. Project the divisions of the tee vertically into the pipe and where they intersect, project them horizontally.
   
   
4. Divide the blank size into 12 equal parts (see Divide a Line Using a Ruler on an Angle). Remember to only divide the circumference, any seam allowance should not be included in the divisions.
   
   
5. Label the stretch-out to match the profiles, starting at the seam.
   
6. Start the pattern at your seam and follow the labeling. Notice the pattern of over 1, up/down 1, until you reach the last line and then it reverses.
   
   
7. Draw the miter line on the pattern with a flexible curve
   

1. Draw a full front elevation view and end elevation view.
2. Draw profiles on the tee in both views and divide into 6 equal parts (see Divide a Circle Into 12 Equal Parts). Include all labeling. Remember that the labeling will flip with the view.
3. Draw in the element lines from the profile divisions.
4. Where the element lines hit the pipe in the end view, project them horizontally into the front view.
5. Following the labeling, draw in the miter line in the front view.
6. Draw the tee stretch-out and divide into 12 equal parts (see Divide a Line Using a Ruler on an Angle). Label the stretch-out to match the elevation view.
7. Transfer/project the corresponding points from the front elevation view to the stretch-out.
8. Circle the points and use a flexible curve to complete the pattern.

1. Draw a full front and end elevation view.
2. Draw profiles on the tee in both views and divide into 6 equal parts (see Divide a Circle Into 12 Equal Parts). Include all labeling.
3. Draw in the element lines from the profile divisions.
4. Where the element lines hit the pipe in the end view, project them horizontally into the front view.
5. Following the labeling, draw in the miter line in the front view.
6. Draw the tee stretch-out to the right and divide it into 12 equal parts (see Divide a Line Using a Ruler on an Angle). Label to match the elevation views.
7. Using your compass, transfer the corresponding points from the front elevation view to the stretch-out.  Note: The element lines cannot be projected in this case because the stretch-out is not perpendicular to the tee. In the case of a 90° tee, it is at a right angle to the stretch-out. To be able to project an oblique tee, the stretch-out must be draw at an angle equal to the tee angle. This usually takes too much room to justify doing so.
8. Use a flexible curve to join the points and complete the pattern.

1. Use the elbow rule (# of pcs × 2 − 2) to find the number of gores. For our example, it is 4 × 2 − 2 = 6. Each end piece is made up of one gore and each middle piece is made up of two gores.
   
2. Use the angle of the elbow divided by the number of gores to find the miter angle. In our case, 90° ÷ 6 = 15°.
3. Now that we know the miter angle is 15°, we can use an end gore and lay it out similar to a “Pipe on a Miter” (see Pipe on a Miter).
   1. Draw an elevation view, complete with the miter line.
      
   2. Draw a profile below (see Divide a Circle Into 12 Equal Parts) and label it.
   3. Project the profile divisions up into the elevation view.
   4. Draw the stretch-out (6 × Pi) directly to the right of the elevation view and divide it into 12 equal parts (see Divide a Line Using a Ruler on an Angle). Label it to match the elevation view and make sure to start the labeling on centre of the gore. Add any required seam allowances outside of the stretch-out.
   5. At the points where the element lines cross the miter line, project them into the stretch-out.
   6. Following the labeling, circle the intersection points on the stretch-out. Don’t put a dot over the points, but circle around them.
   7. Using a flexible curve, join the points to draw in the miter line, completing the pattern.
      
4. We now need to finish the calculation for the blank size of the elbow. We already have the stretch-out, but we need the height. This is found by multiplying the seam height by the number of gores. For our example, simply measure the elevation view and find the height of the element line on centre of the gore. This should be 2 7/16″. 2 7/16″ × 6 =14 5/8″.
5. Now, cut out the blank size.
   
6. Mark the seam height of each gore vertically on the stretch-out.
   
7. To complete the elbow, trace or transfer the first gore pattern onto the blank and cut it out. Then, flip it and trace it for the rest of the gores. Do not flip left to right, only up and down and remember that you must leave 2 seam heights (2 gores) for the middle pieces. This will allow the seams to be orientated on opposing sides and produce the “fish” pattern.
    

1. Draw an elevation view.
   
2. Profile the base of the elevation view and divide it into six equal parts (see Divide a Circle Into 12 Equal Parts).
3. Label the profile from 1 to 7 and project the divisions vertically into the base of the cone.
4. Project the element lines from the base to the apex of the cone.
5. Locate a radius point where you want to develop the pattern. Unlike Parallel line, it doesn’t matter where this is. There is no projection into the pattern like we used before. But keep in mind that you may require enough room to fit a diameter equal to two slant heights.
6. With your compass, take the slant height from the elevation view and swing an arc (stretch-out arc). Because we don’t know how long the arc needs to be yet, we use best judgement. We do know that it will be equal to the base circumference. The shallower the cone is, the larger the stretch-out angle will be. A very steep cone will be a much smaller angle
7. Establish a starting point for the pattern and draw a line back to the radius point. The starting point can be anywhere along the arc
8. Set your compass to a step-off. From your starting point, swing it 12 times along the stretch-out arc.
9. Connect the last point back to the radius point to complete the pattern.

1. Draw a plan and elevation view. When a pyramid is being developed, the plan view must have a point on the X axis to to give it a true length in the elevation view.
   
2. Locate a radius point where you want to develop the pattern.
3. With your compass, take the slant height from the elevation view and swing an arc (stretch-out arc).
4. Establish a starting point for the pattern and draw a line back to the radius point. The starting point can be anywhere along the arc.
5. Set your compass to a length equal to one side of the base. From your starting point, swing it as many times as sides along the stretch-out arc.
6. Connect all the points back to the radius point to complete the pattern.

1. Draw an elevation view, including the apex point.
2. Profile the base of the elevation view and divide it into 6 equal parts.
3. Label the profile from 1 to 7 and project the divisions vertically into the base of the cone.
4. Project the element lines from the base to the apex of the cone.
   
5. Locate a radius point where you want to develop the pattern.
6. With your compass, take the large slant height from the elevation view and swing an arc from the radius point.
7. Set your compass to the small slant height and swing it from the same radius point.
8. Along the stretch-out (large) arc, establish a starting point for the pattern and draw a line back to the radius point.
9. Set your compass to a step-off. From your starting point, swing it 12 times along the stretch-out arc. In this case, this must be done on the large or stretch-out arc because that is where the step-off is taken from.
10. Connect the last point back to the radius point to complete the pattern.

1. Draw an elevation view, including the apex point.
2. Profile the base of the elevation view and divide it into six equal parts.
3. Label the profile from 1 to 7 and project the divisions vertically into the base of the cone.
4. Project the element lines from the base to the apex of the cone.
5. Draw in the miter line.
   
6. Where the element lines cross the miter line, project them horizontally to the outside edge. This now creates seven different slant heights.
   
   
7. Locate a radius point where you want to develop the pattern.
8. With your compass, take the large slant height and swing an arc.
9. Set your compass to all of the remaining slant heights and swing them from the radius point.
10. Establish a starting point for the pattern and draw a line back to the radius point.
11. Set your compass to a step-off. From your starting point, swing it 12 times along the stretch-out arc and label each point to match the elevation view. Start your numbering where you want the seam, commonly put on the short side of the cone.
12. Connect the all of the points back to the radius point.
13. Following your numbering, circle each intersection point. This will create the pattern of over 1, down 1.
14. Use a flexible curve to connect the points and create the pattern.

1. Draw a full plan view complete with all element lines and labeling. Label one half of the plan view, the round end with numbers and the square end with letters. Notice the lines of symmetry in the plan view. Parts of the drawing can be deleted because of this symmetry. In this case, it is on centre in both directions, so a quarter plan view is the minimum required to avoid duplication. But, sometimes it is easier to draw at least a half plan, there is no harm in drawing more than the minimum required.
   
2. Create a true length diagram (TLD) with the vertical height of the fitting and a horizontal length long enough to fit any of the element lines.
3. Take the element lines A-1, A-2 and C-7 from the plan view and place them in the horizontal of the TLD. In this case, all other lines will be a duplicates of these 3 lines. Still, label the TLD with ALL of the element lines so you don’t make a mistake!!!
4. Draw a baseline equal to line A-B. We are now ready to triangulate
5. From the TLD, pick up the true length of line A-4 and swing it upwards from point A towards the centre. Then swing it from point B. Where these arcs cross is point 4. We now have our first triangle, A-B-4. This is our first step of triangulating. Remember that triangulation means using two known points to create a third. In this case, the known points are A and B and the unknown is 4.
6. Next, pick up true length of line A-3 and swing it from point A. In this step our two known points are now points A and 4 (or B and 4) and the unknown is 3. In each step, we will use the last point created as one of our new known points. Because the fitting is symmetrical, continue to work both sides at the same time. We will only discuss one side here, but the steps repeat on the other side.
7. Pick up line 3-4, a step-off, from the plan view and swing it from point 4, to create point 3. When we look at our plan view, we have labeled it in a way that numbers are at one end and letters on the other. So, when we go from one end to the other (number to letter), we need to find the true length, but when we go from number to number (or letter to letter) we don’t have any elevation change involved. We are just going horizontally along the end, which means it is a true length in the plan view.
   
8. Pick up line A-2 and swing it from point A.
9. Pick up a step-off and swing it from point 3, to create point 2
10. Pick up line A-1 and swing it from point A.
11. Pick up a step-off and swing it from 2, to create point 1.
12. Pick up line C-1 and swing it from 1. This is our first step where we are not swinging from A; we now must swing from 1. Always be thinking of the known points and the unknown. We must always swing an arc from a known point. For this triangle, the known points are A and 1 and the unknown is C. So, line C-1 has to be swung from 1 since we don’t know where C is yet.
13. Pick up line A-C from the plan view (remember that letter to letter is true length) and swing it from point A, to create point C. The way to check our work is this last triangle should be a right triangle. Point C should be 90°, if it’s not, go back and check your work.
14. Draw in the all the element lines and outside edges, using a flexible curve for the round end.
15. Cut out and trace the pattern for the other half.

1. Draw a full plan view complete with all element lines and labeling. Label one half of the plan view, travel from large end to small end, zig-zagging back and forth with the numbering 1-14. Notice the lines of symmetry in the plan view. Every round to round will be on centre one way, but it is always worth drawing the whole thing.
2. Create a TLD and label all the element lines. Remember, any element line which travels from one end of the fitting to the other, will need to be put into a true length diagram.
   
3. Draw a vertical line equal to the true length of line 1-2. Square to rounds always start with a horizontal line and round tapers always start with a vertical line. We are now ready to triangulate.
   
4. From the TLD, pick up the true length of line 2-3 and swing it from point 2, back towards point 1. Remember to work both sides at the same time.
5. Next, pick up step-off 1-3 and swing it from point 1, to complete point 3. When we look at our plan view, we have labeled it in a way that odd numbers are at one end and even on the other. So, when we go from one end to the other (odd to even), we need to find the true length, but when we go from even to even or odd to odd we don’t have any elevation change involved. We are just going horizontally along the end, which means it is a true length in the plan view.
6. Pick up line 3-4 from the TLD and swing it from point 3 back towards point 2.
7. Pick up the step-off and swing it from point 2, to complete point 4.
8. Follow this same procedures, swing a true length of an element line and a step-off to create the next point until you reach point 14.
9. Draw in the all the element lines and outside edges, use a flexible curve for the round ends. Because this is a fitting that could be done in radial line, although maybe not practical, the pattern will follow the same shape, having a common radius point and parallel arcs.

1. Draw a full plan view complete with all element lines and labeling. Choose a line of symmetry to place the seam so you only need to make 1 pattern. Label one half of the plan view, the round end with numbers and the square end with letters.
2. Create a TLD and label it
   
3. Draw a baseline equal to line A-B.
   
4. Pick up the true length of line A-4 and swing it from point A.
5. Next, pick up true length of line B-4 and swing it from point B. Where it crosses the first arc, becomes point 4.
6. Pick up a step-off from the plan view and swing it from point 4. Unlike an on-centre square to round, this fitting needs to be developed one side at a time. Complete one side, then go back to finish the other side.
7. Pick up the true length of line A-3 and swing it from point A.
8. Pick a step-off and swing it from point 3.
9. Pick up the true length of line A-2 and swing it from point A.
10. Pick up a step-off and swing it from 2.
    
11. Pick up the true length of line A-1 and swing it from point A.
12. Pick up the true length of line D-1 and swing it from 1.
13. Pick up line A-D from the plan view (remember that letter to letter is true length) and swing it from point A. Remember that point D should be 90°. If it not, go back and check your work.
    
14. Now, go back to point 4 and work the pattern to the other side, completing it at point C. Notice in this example, line C-7 has no length in the plan view, it is a dot. The true length is the vertical height. Zero plan length put 90 deg to the vertical height, will have no change to the vertical height.
15. Draw in the all the element lines and outside edges, using a flexible curve for the round end.

1. Draw a full plan view complete with all element lines and labeling. Label one half of the plan view, go from large end to small end, zigzagging back and forth with the numbering 1-14. Remember, every round to round will be on centre one way, but it is always worth drawing the whole thing.
2. Create a TLD and label all the element lines.
3. Draw a vertical line equal to the true length of line 13-14.
4. From the TLD, pick up the true length of line 13-12 and swing it from point 13 back towards point 14. Remember to work both sides at the same time.
5. Next, pick up step-off 14-12, and swing it from point 14.
6. Pick up line 12-11 from the TLD, and swing it from point 12.
7. Pick up step-off 11-13, and swing it from point 13.
8. Follow this same procedures, swing the true length of an element line and a step-off to create the next point until you reach point 1.
9. Draw in the all the element lines and outside edges, use a flexible curve for the round ends.

1. Draw a full plan view complete with all element lines and labeling.
2. Create a TLD with all element lines required.
3. Start with baseline of two known points then triangulate the third point.
4. Follow the basic steps of triangulation to finish the pattern. In the case of a square to round with NO line of symmetry, two different half patterns will need to be developed.