The Silhouette software is very powerful. In this project, you will learn how to use the software and explore the mathematics behind the creation of shapes.

**Using the drawing tools (oval shape), make a circle with a diameter of one inch.**

Hold down the shift key and drag the circle until it is 1 inch. If you can not get the measurement exact, use the transform window scale to change both the width and height to 1 inch.

**Using the drawing tools (oval shape), make an oval with a height of .05 inch and width of 1 inch.**

Hold down the shift key and drag the oval until it is 1 inch. If you can not get the measurement exact, use the transform window scale to change the height to .05 inch and width to 1 inch.

**Using the drawing tools (pentagon shape), make an equilateral triangle with the length of one inch.**

Move the slider to decrease the number of sides to three. There is a red dot in the middle of one of the sides. Grab the dot and move it downwards while holding down the shift key. The triangle will then have one side which is parallel to the page grid if it is turned on in page setup window. In the Transform tool, go to the scale window to change the size of the triangle to one inch. The lock aspect ratio needs to be locked. Change the width to one and apply the change. Go to the Object menu and click on the convert to path to create a compound figure.

**Using the line tools, make a right triangle.**

Draw a straight line with a length of one inch. Hold down the shift key to get a line which is vertical to the page grid by moving the cursor from top to bottom of the page. In the Transform tool, go to the scale window and change the height of the line to one inch and apply the change. This forms one leg of a right triangle. In the advanced replicate window, with the line highlighted, rotate one copy 90 degrees. Move the copy so that a right angle is formed and the points of each line meet accurately. Zoom in to see that the lines are not crossing one another. The second leg of a right triangle is now formed. With the line that you just created, replicate another copy of the line by rotating this copy 45 degrees. Notice that the line is not long enough. Adjust the line so the bottom right point is touching the horizontal line. Take the upper left box and stretch the line until it meets the vertical line on top. This line is called a hypotenuse. Using the Pythagorean theorem, a

^{2}+ b^{2}= c^{2}, where a and b are the legs and c is the length of the hypotenuse. Is it the same length as the one that you created empirically? If not adjust it. The hypotenuse length should be 1.41 inches if the legs are both 1 inch. Highlight the equilateral triangle, by dragging the cursor over the entire figure. Go to the Object menu, and go to make compound path. Double click your mouse and you will see red dots on the three corners. Click on each dot to make it a gray square (it goes from white to gray) to make a compound figure.**Using the drawing tools (rectangular shape), make a square with a length of one inch.**

Hold down the shift key and drag the square until it is 1 inch. If you can not get the measurement exact, use the transform window scale to change both the width and height to 1 inch.

**Using the drawing tools (rectangular shape), make a rectangle with a width of one inch and a height of two inches.**

Drag the cursor to make the correct width and height. If you can not get the measurement exact, use the transform window scale to change both the width and height.

**Using the drawing tools (pentagon shape), make a pentagon with a side length of one inch.**

The Silhouette software does not give a side length measurement. You need to calculate it using the following formula:

Tangent = Opposite

Adjacent

A pentagon has five sides.

If you divide 360° by 5 sides equals 72° Each triangle formed in the center, has an angle measure of 72°. When dividing the triangle to make a right triangle. The measure of the angle at the center becomes 1/2 of 72° which is 36°.

Using the Tangent formula:

tan(36°) = .5 Where h is the height

h

h = .688 in. (This measure is called the apothem. It can not be doubled here because this is the radius of the incircle.)

To find the height of the rest of the pentagon, you need to use the Pythagorean theorem.

a

^{2}+ b^{2}= c^{2}
a

^{2}+ b^{2}= c^{2}
.25 + .473 = c

^{2}
.723 = c

^{2}
.85 in = c

Adding the two measurements, you will have the height of the pentagon

.688 + .85 = 1.538 in

A regular pentagon with a side length of 1 will have a height of 1.538 inches

**Using the drawing tools (pentagon shape) and adjusting the slider to 6, make a hexagon with a side length of one inch.**

The Silhouette software does not give a side length measurement. You need to calculate it using the same Tangent formula as above substituting six sides. The only difference is that you do not have to use the Pythagorean theorem because the apothem can be doubled here.

A hexagon has six sides.

If you divide 360° by 6 sides equals 60° Each triangle formed in the center, has an angle measure of 60°. When dividing the triangle to make a right triangle. The measure of the angle at the center becomes 1/2 of 60° which is 30°.

Using the Tangent formula:

tan(30°) = ,5 Where h is the height

h

h = .867 in.

.867 + .867 = 1.734 in.

A regular hexagon with a side length of 1 will have a height of 1.734 inches

Repeat the calculations for a heptagon, octagon and decagon. When the number of sides is odd, you need to use the Pythagorean theorem equation in addition to the Tangent formula to add the two values together to get the height of the regular polygon. When the number of sides is even, the apothem can be doubled to obtain the height of the regular polygon.

Save all of these shapes in your Silhouette library. This will become a valuable resource in the future and save you a lot of time when you are designing models. This is similar to a carpenter who has a tool box. It makes designing easier. These are basic shapes which will be used over and over again because they are exactly one inch. They can be combined easily with one another to create a compound model. Don't worry that the size is too small. Make it small and then enlarge the entire model as a whole to make it the correct size once completed.

Disclaimer: there might be instances when enlarging a model will not work. For instance, slice forms need slits that are exactly the thickness of a piece of paper and can not be enlarged. They need to be designed with the correct thickness. In this instance, enlarge the shapes to the correct size and then work on the design.

Save all of these shapes in your Silhouette library. This will become a valuable resource in the future and save you a lot of time when you are designing models. This is similar to a carpenter who has a tool box. It makes designing easier. These are basic shapes which will be used over and over again because they are exactly one inch. They can be combined easily with one another to create a compound model. Don't worry that the size is too small. Make it small and then enlarge the entire model as a whole to make it the correct size once completed.

Disclaimer: there might be instances when enlarging a model will not work. For instance, slice forms need slits that are exactly the thickness of a piece of paper and can not be enlarged. They need to be designed with the correct thickness. In this instance, enlarge the shapes to the correct size and then work on the design.

Awesome calculation method s

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