Making paper gears are an excellent way to see how gears function. It's easy to make different toothed gears with the Silhouette software using the Replicate function. I suggest following my examples first and then making your own with your calculations.
Here is the .Studio file. I am not including a PDF version as these directions are specific to the Silhouette software.
The first thing that you need to do is determine the shape of the tooth. I have included three different styles of teeth. The next thing you need to decide is how many teeth that you want in the gear. I am going to make two gears, one with nine teeth and the other with twelve teeth. The gears need to have the same teeth and distance between the teeth to mesh correctly. My first example will have teeth that are .25 inches wide. Since there are nine teeth, there will be nine spaces in between the teeth. The spaces in between the teeth are also .25 inches wide to accommodate the other gear meshing with this gear. Calculate the circumference of the smaller circle by multiplying the width of the teeth by the number of spaces and teeth. .25 x (9 + 9) = 4.5 inches. The diameter of the circle can be obtained by using the circumference formula.
C = πd
4.5 inches = πd
d = 1.433 inches
The diameter of the circle for the nine toothed gear needs to be 1.433 inches.
Calculate the circumference of the larger circle by multiplying the width of the teeth by the number of spaces and teeth. .25 x (12 + 12) = 6 inches. The diameter of the circle can be obtained by using the circumference formula,.
C = πd
6 inches = πd
d = 1.91 inches
The diameter of the circle for the twelve toothed gear needs to be 1.91 inches.
I have included two circles with the correct measurements in my .Studio file. In the file, there are three tooth designs with a line that protrudes on the left side of the tooth. This appendage is necessary to use the Replicate "grab handle". Highlight the tooth that you want to replicate, in the Replicate window go to the Object on Path window. Click on the "Show Grab Handle". You will see that the circle with the dot is place on the point where the tooth will be placed on the circle. Move the tooth to the top of the circle that is 1.433 inches in diameter. Perpendicular and number of repeats should be checked in the Replicate window. Increase or decrease the number of repeats to 9 for the smaller gear. You will notice that the teeth are not sitting on the circle correctly. Increase the value of the Perpendicular Start Angle until all of the teeth are touching the circle. I had to Zoom in to see that I had to increase the Start Angle to 11. Highlight the gear and the circle. In the Object window, click on the Convert to Path. The grab handle will disappear and the circle will no longer be bold. Highlight the gear and use the Offset window to .01 and then apply. Go to the Object window to release the compound path. Select the outer line which looks like your nine toothed gear and highlight it to Offset the image with an Internal Offset of .01. Discard the outer image by deleting it. The inner image is the nine toothed gear.
Repeat the above instructions for the twelve toothed gear. After you have made the twelve toothed gear, see if you can make your own gears with different number of teeth. Remember, you need to calculate the diameter of the circle because each toothed gear will have different measurements. I have included a tooth with a width of .5 so that you can have different width option too.
Please note: The center of the gear might be skewed a little because of the location of the teeth. I always save the circle that was used to make the gear. Using this circle will allow you to center the axle hole correctly. Once the axle hole is centered in the circle, ungroup both images. Manually move the gear image to center it in this circle with the axle hole. Delete the outer circle leaving the axle hole behind with the gear. Group these two images together to complete the process of making a gear.