Taper worksheet

1. Small end diameter: 1.63 in

1. Find the half angle: each side of the cone leans away from the centerline by half the full (included) angle

   $\frac{7\text{\hspace{0.17em}°}}{2}$$=3.5\text{\hspace{0.17em}°}$

2. Find the small end diameter: each side slopes inward along the length, so take that inward slope off both sides of the large diameter D

   $2-2\times 3\times \text{tan}3.5\text{\hspace{0.17em}°}$$=1.633024\text{\hspace{0.17em}in}$

3. Find the included angle, the full angle of the cone: twice the half angle

   $2\times 3.5$$=7\text{\hspace{0.17em}°}$

4. Find the taper per foot: the taper per inch times the 12 inches in a foot

   $0.12232524\times 12$$=1.467903\text{\hspace{0.17em}in/ft}$

5. Find the tailstock offset (how far to shift the tailstock sideways). Take half the diameter change, then scale it up from the taper length to the whole length between centers, because the whole part tilts

   $\frac{2-1.633024}{2}\times \frac{5}{3}$$=0.305813\text{\hspace{0.17em}in}$

2. Included angle: 10 °

1. Find the taper per inch: how much the diameter shrinks for each unit of length

   $\frac{1-0.65}{2}$$=0.175\text{\hspace{0.17em}in/in}$

2. Find the half angle, how far each side leans from the centerline. Each side drops by half the diameter change over the length, and arctan turns that slope into an angle

   $\text{arctan}\left(\frac{1-0.65}{2\times 2}\right)$$=5.000645\text{\hspace{0.17em}°}$

3. Find the included angle, the full angle of the cone: twice the half angle

   $2\times 5.000645$$=10.001289\text{\hspace{0.17em}°}$

4. Find the taper per foot: the taper per inch times the 12 inches in a foot

   $0.175\times 12$$=2.1\text{\hspace{0.17em}in/ft}$

5. Find the tailstock offset (how far to shift the tailstock sideways). Take half the diameter change, then scale it up from the taper length to the whole length between centers, because the whole part tilts

   $\frac{1-0.65}{2}\times \frac{6}{2}$$=0.525\text{\hspace{0.17em}in}$

3. Included angle: 35.49 °

1. Find the taper per millimeter: how much the diameter shrinks for each unit of length

   $\frac{2-0.72}{2}$$=0.64\text{\hspace{0.17em}mm/mm}$

2. Find the half angle, how far each side leans from the centerline. Each side drops by half the diameter change over the length, and arctan turns that slope into an angle

   $\text{arctan}\left(\frac{2-0.72}{2\times 2}\right)$$=17.744672\text{\hspace{0.17em}°}$

3. Find the included angle, the full angle of the cone: twice the half angle

   $2\times 17.744672$$=35.489343\text{\hspace{0.17em}°}$

4. Find the tailstock offset (how far to shift the tailstock sideways). Take half the diameter change, then scale it up from the taper length to the whole length between centers, because the whole part tilts

   $\frac{2-0.72}{2}\times \frac{6}{2}$$=1.92\text{\hspace{0.17em}mm}$

4. Small end diameter: 0.21 mm

1. Find the half angle: each side of the cone leans away from the centerline by half the full (included) angle

   $\frac{15\text{\hspace{0.17em}°}}{2}$$=7.5\text{\hspace{0.17em}°}$

2. Find the small end diameter: each side slopes inward along the length, so take that inward slope off both sides of the large diameter D

   $1-2\times 3\times \text{tan}7.5\text{\hspace{0.17em}°}$$=0.210085\text{\hspace{0.17em}mm}$

3. Find the included angle, the full angle of the cone: twice the half angle

   $2\times 7.5$$=15\text{\hspace{0.17em}°}$

4. Find the tailstock offset (how far to shift the tailstock sideways). Take half the diameter change, then scale it up from the taper length to the whole length between centers, because the whole part tilts

   $\frac{1-0.210085}{2}\times \frac{5}{3}$$=0.658262\text{\hspace{0.17em}mm}$

5. Small end diameter: 0.47 mm

1. Find the half angle: each side of the cone leans away from the centerline by half the full (included) angle

   $\frac{15\text{\hspace{0.17em}°}}{2}$$=7.5\text{\hspace{0.17em}°}$

2. Find the small end diameter: each side slopes inward along the length, so take that inward slope off both sides of the large diameter D

   $1-2\times 2\times \text{tan}7.5\text{\hspace{0.17em}°}$$=0.47339\text{\hspace{0.17em}mm}$

3. Find the included angle, the full angle of the cone: twice the half angle

   $2\times 7.5$$=15\text{\hspace{0.17em}°}$

4. Find the tailstock offset (how far to shift the tailstock sideways). Take half the diameter change, then scale it up from the taper length to the whole length between centers, because the whole part tilts

   $\frac{1-0.47339}{2}\times \frac{5}{2}$$=0.658262\text{\hspace{0.17em}mm}$

6. Small end diameter: 0.93 mm

1. Find the half angle: each side of the cone leans away from the centerline by half the full (included) angle

   $\frac{2\text{\hspace{0.17em}°}}{2}$$=1\text{\hspace{0.17em}°}$

2. Find the small end diameter: each side slopes inward along the length, so take that inward slope off both sides of the large diameter D

   $1-2\times 2\times \text{tan}1\text{\hspace{0.17em}°}$$=0.93018\text{\hspace{0.17em}mm}$

3. Find the included angle, the full angle of the cone: twice the half angle

   $2\times 1$$=2\text{\hspace{0.17em}°}$

4. Find the tailstock offset (how far to shift the tailstock sideways). Take half the diameter change, then scale it up from the taper length to the whole length between centers, because the whole part tilts

   $\frac{1-0.93018}{2}\times \frac{6}{2}$$=0.10473\text{\hspace{0.17em}mm}$

7. Included angle: 3.82 °

1. Find the taper per inch: how much the diameter shrinks for each unit of length

   $\frac{1-0.8}{3}$$=0.06666667\text{\hspace{0.17em}in/in}$

2. Find the half angle, how far each side leans from the centerline. Each side drops by half the diameter change over the length, and arctan turns that slope into an angle

   $\text{arctan}\left(\frac{1-0.8}{2\times 3}\right)$$=1.909152\text{\hspace{0.17em}°}$

3. Find the included angle, the full angle of the cone: twice the half angle

   $2\times 1.909152$$=3.818305\text{\hspace{0.17em}°}$

4. Find the taper per foot: the taper per inch times the 12 inches in a foot

   $0.06666667\times 12$$=0.8\text{\hspace{0.17em}in/ft}$

5. Find the tailstock offset (how far to shift the tailstock sideways). Take half the diameter change, then scale it up from the taper length to the whole length between centers, because the whole part tilts

   $\frac{1-0.8}{2}\times \frac{3}{3}$$=0.1\text{\hspace{0.17em}in}$

8. Included angle: 0.72 °

1. Find the taper per millimeter: how much the diameter shrinks for each unit of length

   $\frac{1-0.975}{2}$$=0.0125\text{\hspace{0.17em}mm/mm}$

2. Find the half angle, how far each side leans from the centerline. Each side drops by half the diameter change over the length, and arctan turns that slope into an angle

   $\text{arctan}\left(\frac{1-0.975}{2\times 2}\right)$$=0.358094\text{\hspace{0.17em}°}$

3. Find the included angle, the full angle of the cone: twice the half angle

   $2\times 0.358094$$=0.716188\text{\hspace{0.17em}°}$

4. Find the tailstock offset (how far to shift the tailstock sideways). Take half the diameter change, then scale it up from the taper length to the whole length between centers, because the whole part tilts

   $\frac{1-0.975}{2}\times \frac{7}{2}$$=0.04375\text{\hspace{0.17em}mm}$

9. Small end diameter: 0.9 mm

1. Find the half angle: each side of the cone leans away from the centerline by half the full (included) angle

   $\frac{2\text{\hspace{0.17em}°}}{2}$$=1\text{\hspace{0.17em}°}$

2. Find the small end diameter: each side slopes inward along the length, so take that inward slope off both sides of the large diameter D

   $1-2\times 3\times \text{tan}1\text{\hspace{0.17em}°}$$=0.89527\text{\hspace{0.17em}mm}$

3. Find the included angle, the full angle of the cone: twice the half angle

   $2\times 1$$=2\text{\hspace{0.17em}°}$

4. Find the tailstock offset (how far to shift the tailstock sideways). Take half the diameter change, then scale it up from the taper length to the whole length between centers, because the whole part tilts

   $\frac{1-0.89527}{2}\times \frac{5}{3}$$=0.087275\text{\hspace{0.17em}mm}$

10. Included angle: 21.24 °

1. Find the taper per inch: how much the diameter shrinks for each unit of length

   $\frac{2-0.875}{3}$$=0.375\text{\hspace{0.17em}in/in}$

2. Find the half angle, how far each side leans from the centerline. Each side drops by half the diameter change over the length, and arctan turns that slope into an angle

   $\text{arctan}\left(\frac{2-0.875}{2\times 3}\right)$$=10.619655\text{\hspace{0.17em}°}$

3. Find the included angle, the full angle of the cone: twice the half angle

   $2\times 10.619655$$=21.239311\text{\hspace{0.17em}°}$

4. Find the taper per foot: the taper per inch times the 12 inches in a foot

   $0.375\times 12$$=4.5\text{\hspace{0.17em}in/ft}$

5. Find the tailstock offset (how far to shift the tailstock sideways). Take half the diameter change, then scale it up from the taper length to the whole length between centers, because the whole part tilts

   $\frac{2-0.875}{2}\times \frac{6}{3}$$=1.125\text{\hspace{0.17em}in}$