Answer:
11.75, 2.9375
12, 3
12.48, 3.12
Step-by-step explanation:
First, let's go over what they want us to do.
Boring but brief history lesson of geometry:
In the beginning of the history of geometry and circles, mathematicians would find the approximate circumference of circles by drawing polygons inside of circles and finding the perimeter of that polygon. In other words, the approximate circumference of the circle about the perimeter of the shape. The more side lengths there are, the smaller each side is, and the closer the perimeter is to the actual circumference (more on this later)
Problem 1:
The pentagon has a side length of 2.35, so the perimeter is 5 * 2.35 = 11.75
Thus, the approximate circumference is 11.75.
The diameter of a circle is r * 2, or 4, so the ratio of the circumference to the diameter is 11.75 / 4, or 2.9375
Problem 2:
The hexagon has a side length of 2, so the perimeter is 6 * 2 = 12. Thus, the approximate circumference is 12.
The ratio of the circumference to the diameter is 12 / 4, or 3.
Problem 3:
The dodecagon (12 sided shape) has a side length of 1.04, so the perimeter is 12 * 1.04 = 12.48.
The ratio of the circumference to the diameter is 12.48 / 4 = 3.12
Although we are done, let's ~circle~ back (sorry for the pun) to the idea that this exercise is trying to tell us.
The circumference of a circle is 2πr (which is also equal to the diameter times pi). So in this example, the circumference should be 2 * π * 2, or 4π. π is approximately 3.14159, so 4π is about 12.56636. And the ratio of this to the diameter (which is 4) is, if you look at how we got 4π, should be π no matter what.
Note: the definition of π is the ratio of a circles circumference to its diameter, so this makes sense.
Do you see how the approximate circumferences are slowly getting closer to the actual circumference of 12.56?: 11.75, 12, and 12.48
Look at the progression of approximate circumferences / diameter: 2.9375, 3, 3.12. These are slowly getting closer to π (3.14)
As you can see, the more sides, and the more "circular" the shape becomes, the closer the polygon/approximate circumference is to an actual circle.
I hope this helps! Feel free to ask any questions! :)
