Question

Write a formula for the area of the regular polygon. Solve the formula for the height h.

Answer 1

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.

`\begin{gathered} \text{Area of triangle is given by}; \\ A=(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*(1)/(2)bh \\ A=(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* b \\ A=(Ph)/(2) \\ \text{For an octagon, n = 8sides} \\ P=8b \\ \\ \text{Thus,} \\ A=(8bh)/(2) \end{gathered}`

Therefore, the area of the regular polygon (octagon) is;

`A=(8bh)/(2)`

Part B:

To solve for the formula for the height h, we make h the subject of the formula;

`\begin{gathered} A=(8bh)/(2) \\ \text{Cross multiplying:} \\ 2A=8bh \\ \text{Dividing both sides by 8b;} \\ h=(2A)/(8b) \end{gathered}`

Therefore, the height is;

`h=(2A)/(8b)`