In this assignment we will construct cones out of wedged shaped pieces of paper. We will also see how to build a cone of a prescribed radius and height. The radius and height of a cone are the distances labelled r and h, respectively, in the picture above.
To begin, cut out several wedges out of paper. Vary the angle q to get different shaped cones. This will give you some feel for what shape of a cone is built from what size of wedge. By the way, the symbol q is a letter of the Greek alphabet, and is pronounced theta. It is commonly used as a name for an angle.
Next, build a cone of height 6 centimeters and radius 3 centimeters. To do this, you need to know the values of R and q in order to cut out the appropriate wedge. You therefore need to determine how to get R and q in terms of r and h. Here are two things to keep in mind to help you do this. First, you can get l from r and h by the Pythagorean Theorem by noticing that r, h, and l form a right triangle. Next, the circumference of the wedge is the part of the circumference of the circle that is on the wedge. The relationship between the angle q and this circumference is that
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In other words, the faction of the total circumference taken up by the wedge is the same as the fraction off the entire 360 degree angle that q makes. From this formula, along with the formula for the circumference of a circle, we see that the circumference of the wedge is 2pR(q / 360).
You should also ask yourself what is the relationship between R and l and also what is the relationship between the circumference of the wedge and the circumference of the cone. It is perhaps easiest to find these relationships by looking carefully at how you take a wedge and turn it into a cone.
Problem. Describe clearly, with the aid of appropriate pictures, how you built a cone of radius 3 cm and height 6 cm. Make sure to say to how you determined R and the radius of the wedge, both in terms of the finished cone. Draw the wedge you used to build the cone.