Answer
a. $2$
b. $2.8284$
c. $3.0615$
d. all are smaller than the true value.
Work Step by Step
Step 1. As shown in the diagram, for a n-sided polygon, each isosceles triangle will have an angle of $\theta=\frac{2\pi}{n}$
Step 2. With a unit circle, the area of each small triangle is given by $A_n=\frac{1}{2}(base)(height)=\frac{1}{2}(2sin\frac{\theta}{2})(cos\frac{\theta}{2})=\frac{1}{2}sin\theta=\frac{1}{2}sin(\frac{2\pi}{n})$
Step 3. The total area will be the sum of all small triangles: $A=\sum A_n =\frac{n}{2}sin(\frac{2\pi}{n})$
a. For $n=4$, we have $A(4)=\frac{4}{2}sin(\frac{2\pi}{4})=2$
b. For $n=8$, we have $A(8)=\frac{8}{2}sin(\frac{2\pi}{8})=2\sqrt 2\approx2.8284$
c. For $n=16$, we have $A(16)=\frac{16}{2}sin(\frac{2\pi}{16})=8sin(\frac{\pi}{8})\approx3.0615$
d. The precise area of the circle is $A_0=\pi r^2=\pi=3.14159$. Thus all the results from parts a-c are less than the actual value, but the results get closer to the true value.
