Determine the area of a regular octagon with vertices on the unit circle

Question

How would I determine the area? Help please

Answer 1

Note that a regular octagon looks like this (Wikipedia):

So, the triangles in the diagram are isosceles with side length $1$, and their area is given by $\frac{1}{2} (1)^2 \sin (45^\circ)$. There are 8 such triangles, so multiply this by $8$ to get the overall area.

Answer 2

Notice, in general the area $(A_n)$ of a regular n-sided polygon whose vertices are on a circle having radius $R$ is given as can be easily derived as $$A_n=n\left(2\times \frac{1}{2}\times R\sin\frac{\pi}{n}\times R\cos\frac{\pi}{n}\right)$$

$$\bbox[5px, border:2px solid #C0A000]{\color{blue}{A_n=\frac{1}{2}nR^2\sin\left(\frac{2\pi}{n}\right)}}$$ $\forall\ \ \ n\ge 3\ \ \ (n\in N)$

Hence, for a regular octagon $n=8$ & radius of unit circle $R=1$, one should get area of regular octagon as $$A=\frac{1}{2}(8)(1)^2\sin\left(\frac{2\pi}{8}\right)=4\frac{1}{\sqrt 2}=\color{red}{2\sqrt 2}$$