Solution:
Part A:
The image of the regular polygon given is an octagon.
An octagon is a polygon with 8 sides.
To calculate the area of a regular polygon, the polygon is split into triangles and the area of triangles is summed up to get the area of the polygon.
[tex]\begin{gathered} \text{Area of triangle is given by}; \\ A=\frac{1}{2}bh \\ \text{where b is the base} \\ h\text{ is the height.} \\ \\ A\text{ polygon has n-triangles.} \\ \text{Therefore, the area of a regular polygon is;} \\ A=n\times\frac{1}{2}bh \\ A=\frac{n}{2}bh \\ \\ \text{Also, the perimeter of the polygon is the sum of the outer sides, i.e, the sum of the base.} \\ P=n\times b \\ A=\frac{Ph}{2} \\ \text{For an octagon, n = 8sides} \\ P=8b \\ \\ \text{Thus,} \\ A=\frac{8bh}{2} \end{gathered}[/tex]
Therefore, the area of the regular polygon (octagon) is;
[tex]A=\frac{8bh}{2}[/tex]
Part B:
To solve for the formula for the height h, we make h the subject of the formula;
[tex]\begin{gathered} A=\frac{8bh}{2} \\ \text{Cross multiplying:} \\ 2A=8bh \\ \text{Dividing both sides by 8b;} \\ h=\frac{2A}{8b} \end{gathered}[/tex]
Therefore, the height is;
[tex]h=\frac{2A}{8b}[/tex]
